CALIBRATION OF TWO MODELS FOR ESTIMATING REFERENCE EVAPOTRANSPIRATION BY USING FAO-56 PENMAN-MONTEITH MODEL UNDER ARID CONDITIONS

The Penman-Monteith method (P-M) to estimate the reference evapotranspiration (ET o ) is the most reliable method and recommended by the FAO as the standard to verify other empirical methods. However, the Thornthwaite (Th) and Hargreaves-Samani (H-S) models are widely used because they are based on measurements of air temperature, frequently recorded in any meteorological stations. In this study, the daily meteorological parameters of air temperature, relative humidity, wind velocity, were available at six stations (Riyadh), (Ha’il), (Tabuk), (Turayf), (Makkah) and (Jazan). The net radiation was computed using a mathematical model based on a serie of related equations. Therefore, the application of Penman-Monteith became possible to calibrate the Thornthwaite and Hargreaves-Samani models. The local calibration of the both models (Th and H-S) in arid conditions is based on modifying the original coefficients of the named models using the ratio for estimated ET (Th and H-S mpdels) and the reference ET o of (P-M model). In the comparison, the indices of concordance (D), confidence (C), correlation coefficient (r) were analyzed, together with the root mean square error (RMSE) and Nash-Stucliff Efficiency (NSE). So, the ET of H-S model without adjustment were greater than the ET o of P-M during all the months at the total of the studied stations. Contrary, the use of non-adjusted Th ET show a smaller values of the monthly average in a total of the selected stations. After adjustment of the original coefficients of (0.0023) for H-S model and (1.6) for the Th model, we can obtain the new equations of estimating the monthly average of ET fitting better with the P-M Et o model.


INTRODUCTION
In the countries located in arid zones, the information about evapotranspiration (ET) is significant for water resources planning and human uses (Slabbers, 1977;Stefano and Ferro, 1997;Wu, 1997;Qiu et al., 2013;Benli et al., 2006).Also, ET is very important for understanding the spatial distribution of the natural plant communities (Monteith, 1964(Monteith, , 1965;;DeVries et al., 2010).The knowledge and measurement of the changes in ET are necessary to understand any modification in the energy balance and the eco-hydrological changes (Kabenge et al., 2013;Kosugi and Katsuyama, 2007;Schume et al., 2005;Djaman and Irmak, 2013a;Djaman et al., 2013b;Irmak et al., 2013;Rijsberman and Frank, 2006).Potential ET is a related to the atmospheric forcing and surface types.In general, techniques for estimating ETo are based on one or more atmospheric variables, or on some measurements related to these variables, like pan evaporation (Bautista and Bautista, 2009).
The Penman-Monteith method (PM) is considered to be the most physical model and recommended by the FAO as the sole standard to verify other empirical methods (Allen et al., 1998).The FAO Penman-Monteith method is based on a strong theoretical basis, using the energy balances to model ETo.However, it needs four meteorological parameters: Air temperature, Relative humidity, Wind, and Net radiation, which may not be everywhere available (Smith et al., 1991;Camargo and Camargo, 2000).Other methods as for example, Thornthwaite and Hargreaves require few meteorological data (Thornthwaite, 1948).So, these methods were developed for use in climate studies and are most applied to various climates similar to that where they were developed (Wilson et al., 2001;Berti et al., 2014;Bogawski and Bednorz, 2014;Manoj and Dholakia, 2013).In fact, these methods need recalibrating the constants involved in their formulae in order to be extrapolated to other climatic areas (Valiantzas, 2013;Ravanazzi et al., 2012;Rojas and Schiffield, 2013).Thornthwaite's model (TM) was developed the in east-central USA, based on mean air temperature, a widely available variable, and two tabular indexes: number of sunny hours, and monthly heat index (Jensen, 1973).So, the TM is not recommended for use in areas that are not climatically similar to the east-central USA.
In the hand, Hargreaves' model (HM) is a simple model based only on two meteorological parameters, temperature (mean, maximum and minimum) and incident radiation (Hargreaves and Samani, 1985).Although the incident radiation uses the extraterrestrial radiation (Ra) to estimate the ETo, for a given latitude and day.Ra can be obtained from tables or it can be calculated by means of a set of equations using temperature.In many and various regions, the meteorological stations do not have enough data to use PM.Therefore, many studies aim to develop statistical procedures for estimating regional and temporal adjustments to HM and TM in order to obtain the best estimations of ETo in arid regions (El-Nesr et al., 2010;Jabloun and Sahli, 2008;Mohawesh, 2011;Sabzipavar and Tabari, 2010).
In this study, Saudi Arabia area contains 25 meteorological stations have historical data for calibrating the TM and HM using the Penman-Monteith ETo.The objectives of this study were to evaluate the efficiency of the (TM and HM) equations using the PM model in arid climate.So, the kind of this study can be justified by the fact that temperature-based evaporation calculation methods, although widely criticized, are still widely used in climate studies in Saudi Arabia.

Study Area
The daily data of the year 2017 were the recent available in 25 meteorological stations located in contrasting environmental conditions of various regions in Saudi Arabia.The selected meteorological stations are supervised by the General Authority of Meteorology and Environmental Protection (GAMEP) (Table 1).The geographic locations of the meteorological stations over Saudi Arabia are shown in Figure 1.

Materials and Dataset
In this study, the daily meteorological data recorded during 2017 at six Meteorological stations supervised by the General Authority of Meteorology and Environmental Protection (GAMEP) have been used (Table 3).The selected stations represent the diversity of Saudia Arabia relief.

Penman-Monteith Model (P-M)
The estimation of ETo with Penman-Monteith model uses the following equation: (1) Where Δ is the slope of the saturation vapor pressure versus air temperature curve (kPa °C-1 ); Rn is the daily net radiation (MJm -2 d -1 ); G is the soil heat flux (MJm -2 d -1 ); γ is the psychrometric constant (kPa°C -1 ); T is the daily mean temperature of the air at 2 m of height (°C); U2 is the daily mean of wind speed at 2 m of height (m s -1 ); es is the saturation vapour pressure (kPa); ea is the actual vapour pressure (kPa) (Allen et al., 1998).
The parameters of Eq. ( 1) can be estimated from the observed climatic variables.But, the missing climatic data of Net Solar radiation can be estimated empirically.The Net radiation (Rn) is computed as the algebraic sum of the net short and long wave radiation (Rns and Rnl, respectively).
Rn = Rns -Rnl (2) Where, Rns results from the balance between incoming and reflected solar radiation (Rs) adopting an albedo of 0.23 as follows: Rns = (1 -α) Rs (3) Rs is not measured, it can be estimated from the observed duration of sunshine hours with the Angstrôm equation (Angstrôm, 1924): (4 where Rs is solar or shortwave radiation (MJ m −2 day −1 ), n is actual sunshine duration (h).It can be computed by the following equation: ( where ωs is the sunset hour angle, given by: where Ø is the latitude angle in [rad] and δ is the solar decimation in [rad], computed by: (7) where J is the number of the day in the year between 1 (1 January) and 365 or 366 (31 December).
N is the maximum possible sunshine duration (h).It can be computed by the following equation: (8) n/N is relative sunshine duration, Ra is extraterrestrial radiation (MJ m −2 day −1 ), computed for any given day as a function of the latitude of the site as follows: (9 where Gsc is the solar constant equals 0.0820 MJ m -2 min -1 and dr is the inverse relative distance Earth-Sun given by: (10) Rnl results from the balance between the down-coming and the outgoing long wave radiation emitted by the vegetation and the soil.Computations were performed as proposed by (Allen et al., 1998). where:

Reference Evapotranspiration Estimation Models
Table 2 summarized the selected ET models and the required climatic variables for every model.Tj': monthly average of temperature ( o C), T': daily average of temperature ( o C), Tx: daily average of maximum temperature ( o C), Tm: daily average of minimum temperature ( o C), Ra: extraterrestrial radiation (MJm -2 day -1 ), Rs: solar radiation (MJm -2 day -1 ).

Hargreaves-Samani Model (H-S)
The estimation of ET with Hargeaves-Samani model uses the following equation: The HS method requires only observed Tmin and Tmax for the estimation of ETo (mm day −1 ), which is given as (Hargreaves and Samani, 1985): where: ETo, daily evapotranspiration (mm day -1 ), Tmax, Tmin, T' are the daily maximum, minimum and mean air temperature ( o C), respectively, Ci = 0.002 is the original empirical constant proposed by Hargreaves and Samani, Ra, the water equivalent of the extraterrestrial radiation (mm day -1 ) (Hargreaves and Samani, 1985).

Thornthwaite model (Th)
Thornthwaite analyzed this effect of air temperature on transports water from the earth to atmosphere with data of evapotranspiration (Thornthwaite, 1948).He found that a general form of the relation can be: e = Ct a ; where e, is the monthly evapotranspiration in centimeters and (t) is the mean air temperature in °C.The coefficients (C) and (a) vary from place and season.Thornthwaite proposed a general equation with a value of C = 16.Since the calculation of the evapotranspiration is not appropriate in areas with a monthly average temperature less than 0, an equation was developed to integrate this parameter, which corresponds to the monthly (i) and annual (I) heat index (Bautista et al., 2009).Based on these, he proposed calculating the exponent (a).An adjustment factor relating to the specific number of days per month and hours of sunlight, depending on the season and latitude is integrated to the equation.

Model Performance Evaluation
The performance models was evaluated using five statistical indicators, which are : (Table 3).

Comparison of P-M Eto with Two Models Without Adjustment
From table 4 and the figure 2, the Penman-Monteith ETo exceed those estimated by the Tornthwaite model during the different months, with a difference between 0.7% and 155.9% in the Riyadh and between 55.1% and 165.1% in the Jazan.The P-M ETo also exceed their estimated counterparts during the various months of the year at Hail and Turayf, except for the winter months (December-January-February).During these months, the difference between the estimated and reference values ranged between 4.7% and 17.7% at Hail and between 5.1% and 49.4% at Turayf.During the various months of the year the P-M ETo values exceed their estimated counterparts at Tabuk, except for the month of January, which is characterized by a monthly average greater with 6.8%.At the Makkah station, it also exceeds All P-M ETo exceed the estimated during the various months of the year, except for December, which is characterized by a monthly average greater with 9.7% than the P-M model.Contrary to the Tornthwaite estimates, all the estimated values of the HS model are greater than the P-M ETo during the different months of the year for all stations with a difference ranging between 53.5% (January) and 59.8% (July) in Riyadh, 70.3% (July) and 70.4% (December) in Hail, 47.0% (January) and 71.9% (June) in Tabuk, 46.5% (January) and 63.9% (July) in Turaif, 52.2% (January) and 50.0%(April) in Jizan, and between 59.7% (February) and 72.6% % (June) in Makkah.
The evapotranspiration estimates from both models (PM vs. HM and PM vs. TM) were compared using simple error analysis.The models were compared before and after adjustment.For each location, the following parameters were calculated: Nash-Stucliff Efficiency (NSE), Concordance index (D).Correlation coefficient and RMSE-observations standard deviation ratio model (RSR).Table 5 summarizes the performance results.
The linear regression models between PM and Thornthwaite without adjustment present a negative correlation at Tabuk and the lower correlation values of (r), ranged from 0.20 to 0.47 for the stations of Ha'il, Riyadh and Makkah.But the correlation for Turayf and Jazan are excellent with (0.86 < r < 0.89) respectively.The low values of (r) can be explained by the slope and the value of origin in the y-axis of Thornthwaite ET (Figure 3).For this reason, the confidence index (C) values are also low and ranged from 0.07 to 0.14 , for Ha'il and Turayf, respectively.In the same context, the (1.1 < RSR < 1.7), (-0.1 < NSE < -0.7) and (0.16 < D < 0.66), the (RSR), (NSE) and (D) parameters are unsatisfactory for the named stations.
With HS model, we obtained over-estimations of ET during all the months in total of selected stations.The linear regression models of the monthly relationship between PM and Hargreaves-Samani without adjustment are higher and better than obtained between PM and Thornthwaite models (Table 5 and Figure 4).However, in meteorological stations tested, the Thornthwaite model had a lower "D" and "C" index than that of H-S and P-M models.So, the equation of Thornthwaite has been one of the most misused empirical equations generating inacurrate estimates of evapotranspiration for arid and semiarid areas (Bautista and Bautista, 2009).In dry climatic conditions, many authors concluded that the Thornthwaite model gives the unreliable results of the ET estimations (Chen et al., 2005;Hashimi and Habiban, 1979).So, many researchers propose the modified Thornthwaite equation to improve the performance model in different climatic conditions (Pereira and Pruitt, 2004).Consequently, the ET over-estimations of H-S model and the ET underestimations of Thornthwaite model are modified with changing the values of the original constants of model equations.

Determination of The Adjusted Constants for HM And TM
The test estimations of ETo with Hargreaves-Samani and Thornthwaite were adjusted to the result of the reference equation, that is Penman-Monteith.The determination of the new values of the constants (Ci) of Hargreaves-Samani (eq.16) and Thornthwaite (eq.17) were calculated for each month respectively.Table 6 summarized the mean of the adjusted constants in every meteorological station.
The adjusted constant is obtained by the ratio of the estimated ET of Thornthwaite and Hagreaves-Samani models and the ETo of Penman-Monteith model using the equations (eq.16 & eq.17).For the Thornthwaite model all the adjusted constant values is smaller than the original constant (1.6) from February to November at Riyadh and Tabuk, March to November at Ha'il, March to October at Turayf, January to November at Makkah and during the year at Jazan.The value of the constant of adjusted HM is greater than the value of the original constant (0.0023) in the total of the months and in the total of studied stations.So, the proposed values of the adjusted constant are ranged from 0.0049 to 0.006 fo Riyadh, 0.0045 to 0.0094 for Ha'il, 0.004 to 0.0082 for Tabuk, 0.0046 to 0.0064 for Turayf, 0.0057 to 0.0094 for Makkah and from 0.0037 to 0.0093 for Jazan.

Comparison of PM Eto with Adjusted Models
After adjusting the original constants of both models Thornthwaite and Hargeaves-Samani, the adjusted estimations of ET were summarized in the table 7.Many statistical methods are used to compare the predicted and observed estimations (Efhimiou et al., 2013).In this study, four types of measures are applied to assess the performance of the adjusted ET estimations of Thornthwaite and Hargreaves models using Penman-Monteith ETo: Nash-Stucliff Efficiency (NSE), Concordance index (D).
Correlation coefficient and RMSE-observations standard deviation ratio model (RSR).Table 8 summarizes the performance results.
From table 7, the new monthly adjusted constants exceed the original value (1.6) for the Yhornthwaite model, during all the months, except December and January in Turayf station.While, the adjusted values of the H-S model are smaller than the original value (0.0023) for all the months in the total studied stations.After adjusting the value of the original constants, the performance efficiency of both models Thornthwaite and Hargreaves-Samani was significantly improved (Table 8).
Consequently, we propose the new models equations using the adjusted constant deriving from the average of the monthly constant (Table 10).

CONCLUSIONS
The ET estimations using Thornthwaite and H-S non adjusted equations showed that the two models are not suitable for arid study area.Consequently, the ET estimates of H-S model are greater than the ETo P-M model over the study area.The difference between the estimates of the two models varying from 53 to 62% at Riyadh, 66 to 72% at Ha'il, 48 to 72% at Tabuk, 46 to 64% at Turayf, 60 to 76% at Makkah and from 43 to 53% at Jazan.In the other hand, the ET estimations using Thornthwaite are smaller than the ETo P-M model with a difference ranged from 1 to 61% at Riyadh, 11 to 40% at Ha'il, 1 to 42% at Tabuk, 11 to 52% at Turayf, 14 to 46% at Makkah and from 36 to 62 at Jazan.During the winter season, the ET estimates of Thornthwaite model exceed than the ETo reference of P-M model at Turayf and Ha'il; during December at Makkah and during January at Tabuk.The local calibration of ET estimates showed the possibility of modifying the original coefficients of Thornthwaite and H-S models.The two adjusted models can be suitable alternative to P-M model, which requires some climatic measurements that are not readily available in any meteorological station.So, this study conclude with six alternative equations for estimating the ET using the adjusted models of Thornwaite and H-S.

Figure 1 :
Figure 1: The spatial distribution of the meteorology stations.
ETP: monthly evapotranspiration (cm).I : sum of the monthly thermal Index, ij = (T'/5) 1.514 T' : monthly mean temperature ( o C), α = 0.49239 + (1792 × 10 -5 ij) -(771 × 10 -7 i 2 ) + (675 × 10 -9 i 3 ) j j b : reduction factor, given by the following equation : latitude.yo:tabulated value of reduction factor corresponds to the month and the latitude N preceding the station latitude.y1:tabulated value of reduction factor corresponds to the month and the latitude S next station latitude Xo: latitude N preceding the station site.X1 : latitude S next the station site.2.4 Adjustment of ModelsThe test estimations of ETo with Hargreaves-Samani H-SM, Thornthwaite (TM), Schendel, Ivanov and Blaney-Criddle models were adjusted to the result of the reference equation, that is Penman-Monteith (P-MM).Adjustments were made by changing the value of the corresponding constant, (Ci) in the case of Hargreaves-Samani, (C) in the case of Thornthwaite, (k) in the case of Blaney-Criddle and (a) in the cases of Ivanov and Schendel, with the original values of 0.0023, 16, 0.85 and 0.45, 0.0018 and 16 respectively(Borges and Mendiondo, 2007).The adjusted values of the ET coefficients can be computed as follows: (a of observations, Oi : the value of Penman-Montetith model (PM), O' : the mean of Penman-Monteith model (PM), Pi : the value of the tested models Hargreaves-Samani (H-S).andThornthwaite (Th).

Table 1 :
Meteorology Stations Distributed by Administrative Regions.

Table 3 :
Statistical Indicators Used for The Performance Evaluation of Eto Models.

Table 4 :
Monthly Average of Estimated ET and Reference Eto at The Selected Stations.

Table 6 :
Adjusted Constants of Original Equations of The Thornthwaite (Th) and Hargreaves-Samani (HS) Models.

Table 8 :
Performance Tests of the Adjusted ET Estimations

Table 10 :
The Adjusted Models of Thornthwaite and Hargreaves-Samani.