evaluation of different methods to estimate monthly ... · evaporation or evapotranspiration. among...

Water Utility Journal 18: 61-77, 2018. © 2018 E.W. Publications

Evaluation of different methods to estimate monthly reference evapotranspiration in a Mediterranean area

V.Z. Antonopoulos* and A.V. Antonopoulos School of Agriculture, Dept of Hydraulics, Soil Science and Agricultural Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece * e-mail: [email protected]

Abstract: Many empirical equations and methods have been used and proposed in order to estimate the reference evapotranspiration from land or the evaporation from a free water surface. These empirical methods belong in one of the four categories of a) radiation and temperature b) temperature, c) mass transfer and d) regression or simplified based methods. In this article, thirteen of those empirical methods and a model based on artificial neural networks (ANN) were evaluated and compared with the Penman-Monteith equation, which is considered as a standard method. A meteorological dataset that consisted of monthly meteorological measurements from a station in northern Greece, covering a period of seven years (2009 to 2015) was used. The radiation and temperature based methods of Penman, Priestley-Taylor, Makkink, de Bruin-Keijman, and the modified Penman correlated better with the Penman-Monteith method. The temperature based methods of Hargreaves and Blaney-Criddle followed in accuracy, while the mass transfer equation, the Jensen-Haise, the Copais and the Valiantzas simplified methods presented the lowest correlation. A multilinear regression model based on the meteorological dataset gave highly correlated results. The recalibration of the constant values on some of these equations to the local meteorological conditions showed significant improvement.

Key words: Reference evapotranspiration; Penman–Monteith method; empirical equations; ANNs model; monthly datasets; methods comparison; limited weather data

1. INTRODUCTION

The crop water requirements, which are essential in irrigation planning and scheduling, the hydrologic balance studies and the watershed hydrology, are based on the accurate estimation of the reference evapotranspiration (ETo). Numerous methods or equations have been developed and are being used to estimate the ETo. Those depend mainly upon the availability of meteorological variables. These may be categorized into one of the following classes of energy budget, aerodynamic transfer (or mass transfer), a combination of those and empirical methods.

The suitability of various evapotranspiration methods has been a serious subject for many hydrologists and agriculturists. Evaporation researches that included comparisons of a number of equations were the Winter et al. (1995) and Rasmussen et al. (1995) studies of lakes in the continental climate of Minnesota, the Singh and Xu (1997) comparison of thirteen mass-transfer methods applied to four sites in Ontario, Canada, the Dalton et al. (2004) comparison of evaporation methods for Lake Seminole in Georgia, the Yao (2009) evaluation for the lake evaporation for Dickie Lake, Canada and the evaluation of 14 alternate evaporation methods that were compared with values from the Bowen-ratio energy-budget method by Rosenberry et al. (2007). Antonopoulos et al. (2016) presented an evaluation between different empirical methods of lake evaporation. Evapotranspiration research also includes comparison of numerous equations describing evaporation or evapotranspiration. Among the articles on comparison between different approaches is the work of Xu and Singh (2002), which evaluated five empirical equation of ETo estimation with daily datasets from Switzerland. Rosenberry et al. (2004) compared evapotranspiration rates determined with several empirical methods for a wetland in semi-arid North Dakota. Lu et al. (2005) studied six ETo methods using databases from forested watersheds in the southeastern United States. Xystrakis and Matzarakis (2011), using daily meteorological data from the island of

62 V.Z. Antonopoulos & A.V. Antonopoulos

Crete (southern Greece), evaluated 13 empirical equations of ETo. Rácz et al. (2013) presented a comparative analysis of different methods of ETo for the continental climate of Eastern Hungary. Djaman et al. (2015) evaluated 16 ETo equations against the ASCE-PM equation under the Sahelian conditions in the Senegal River Valley and Antonopoulos and Antonopoulos (2017) evaluated the daily ETo estimates by artificial neural network models and empirical equations using limited input climate variables.

During the last decades there has been a widespread interest in the application of artificial intelligence techniques in the field of water sciences and technologies. Artificial neural networks (ANN) were used to estimate evaporation from free water surface as well as actual and reference evapotranspiration (Jain et al., 2008; Diamantopoulou et al., 2011; Laaboudi et al., 2012; Heddam, 2014; Antonopoulos et al., 2016; Antonopoulos and Antonopoulos, 2017).

Significant efforts for the accurate estimation of ETo under the Greek environmental-meteorological conditions have been carried out by several researchers, which focused on comparisons and sensitivity analysis between the aforementioned methods, their parameters and their calculation time step (Tsakiris and Vangelis, 2005; Sakellariou-Makrantonaki and Vagenas, 2006; Ampas et al., 2007; Aschonitis et al., 2012; Paraskevas et al., 2013; Kitsara et al., 2015; Efthimiou et al., 2013). Additional studies have been performed to develop new methodologies for the estimation of ETo using empirical equations, such as the “Copais model” (Alexandris et al., 2006), equations based on Penman’s formula with reduced parameters (Valiantzas, 2006) or with indirect estimation using pan evaporation measurements and pan coefficients (kp) adapted to the surrounding environmental conditions (Aschonitis et al., 2012).

The aim of this study is to evaluate the applicability and validity of different empirical methods from different categories of input variables as of radiation and temperature, temperature only, mass transfer, based on regression processes or simplification of others to estimate monthly reference evapotranspiration. Models of Artificial Neural Networks technique are also examined and evaluated. Monthly data from a meteorological station in northern Greece was used. The results were compared to the values of reference evapotranspiration estimated by the Penman – Monteith method.

2. MATERIAL AND METHODS

2.1 Monthly reference crop evapotranspiration

The Food and Agricultural Organization of the United Nations (FAO) accepted the FAO Penman-Monteith as the standard equation for the estimation of ET (Jensen et al. 1990; Allen et al., 1998). This method is a combination of the energy balance and the aerodynamic processes and two more resistance factors, the aerodynamic and the (bulk) surface resistances. The surface conductance/resistance term that accounted for the response of leaf stomata to their hydrologic environment was introduced by Monteith (1965).

The FAO-56 Penman-Monteith (PM) method of daily or monthly reference crop evapotranspiration ΕΤο estimation is described by the following equation (Allen et al. 1998):

ΕΤο =0.408Δ(Rn −G)+ γ

900Τ + 273

u2 (eα − ed )

Δ + γ (1+ 0.34u2 ) (1)

where ΕΤο is the daily reference crop evapotranspiration (mm d-1), Rn is the net radiation (MJ m-2d-

1), u2 is the mean wind speed at 2 m above soil surface (m s-1), T is the mean air temperature (oC), G is the soil heat flux density at the soil surface (MJ m-2d-1), ea is the saturation vapour pressure (kPa), ed is the actual vapour pressure (kPa), Δ is the slope of the saturation vapour pressure-temperature curve (kPaoC-1), γ is the psychrometric constant (kPaoC-1).

Water Utility Journal 18 (2018) 63

Many other formulas have been developed which are simplified methods of the Penman formula or other empirical equations, and are grouped according to method type (Winter and Rosenberry, 1995; Rosenberry et al., 2007; Yao, 2009). In this study, a) the combination Penman method, b) the radiation and temperature based methods of Priestley-Taylor, de Bruin-Keijman and FAO-24 Makkink, c) the temperature based methods of Hargreaves and Blaney-Criddle, d) the mass transfer based method and e) some simplified and regression based methods were evaluated and compared with Penman – Monteith method. An artificial neural network is also evaluated and compared with PM method.

2.2 Penman method

The Penman method (Doorenbos and Pruitt, 1977) is a modification of the initial Penman formula, which has the following form:

)ee)(u536.01(62.2G

da2w

−++γ+

λρ−

+=

γΔγΔΔ R

ETo)( n

(2)

The Penman method has been used to estimate the reference crop evapotranspiration in many studies (Yao, 2009; Rosenberry et al., 2007; Antonopoulos et al., 2016).

A modified Penman equation presented by Doorembos and Pruitt (1977) was expresses as

EToPM = cEToP (3)

where EToP and EToPM are the reference evapotranspiration of Penman and the modified equation, respectively and c is an adjustment factor that incorporates the differences between day and night weather conditions. Frevert et al. (1983) and Kotsopoulos and Babajimopoulos (1997) presented (among others) expressions of c as a polynomial of Rs (solar radiation), RHmax, Uday (wind velocity during the day hours) and Ur (the ration of day to night hours velocity). These polynomial expressions of c approach the tabular values given by Doorenbos and Pruitt (1977) with maximum absolute errors of 26.2% and 5.6%, respectively (Kotsopoulos and Babajimopoulos, 1997).

2.3 Priestley-Taylor, de Bruin -Keijman and FAO-24 Makkink methods

The Priestley-Taylor method (Priestley and Taylor, 1972), the de Bruin – Keijman method (de Bruin – Keijman, 1979) and the Makkink (1957) are modifications and simplifications of the Penman formula (Doorenbos and Pruitt, 1977). Evapotranspiration is estimated as a function only from the energy term of the Penman equation. The aerodynamic term is approximated as a fixed fraction of the total evaporation over a suitable average period. The equations have the following form, respectively:

λρ−

+=

w

G)( nRETγΔ

Δαo

(4)

λρ−

+=

w

G63.0

)( nRETγ0.85Δ

Δo

(5)

λρΔ+α=γ+Δ w

smmo

RbET

(6)


where λ is the latent heat of vaporization (MJ kg−1), ρw is the water density (kg m-3) and α is an empirically derived parameter with an average value of 1.26, αm is a coefficient (mm d-1), bm is a nondimensional constant calculated by a function of mean relative humidity (%) and wind speed (m s-1), Rs is solar radiation (MJ m-2 d-1). The values of αm and bm of Makkink method are -0.12 and 0.61, respectively. The Priestley-Taylor, de Bruin – Keijman and Makkink methods have been used to estimate the reference crop evapotranspiration in many works (Utset et al., 2004; Rosenberry et al., 2007; Gianniou and Antonopoulos, 2007; Yao, 2009; Bogawski and Bednorz, 2014; Aschonitis et al., 2015).

2.4 Hargreaves method

The Hargreaves method (Hargreaves and Samani, 1985) estimates daily ETo, when solar radiation, relative humidity and wind speed data are missing by using only the maximum and minimum air temperature in the following equation

( )( ) h

h mean h max minc

o aET a T b T T R= + − (7)

where Tmax, Tmin and Tmean are the maximum, minimum and mean temperature (oC), respectively, Ra is the extraterrestrial radiation (mm day-1) and ah=0.0023 oC-1.5, bh=17.8 oC, and ch=0.5 empirical constants of Hargreaves equation. Hargreaves and Samani (1982) calculated solar radiation (Rs) as:

α−= R)TT(162.0 5.0minmaxsR (8)

The units of constant 0.162 of Eq. (8) are in (oC)-0.5.

2.5 Blaney-Criddle Method

The Blaney-Criddle method (Blaney-Criddle, 1950, Doorembos – Pruitt, 1977; Allen and Pruitt, 1986) formula is:

))13.8T46.0(p(ba meanBCBC ++=oET (9)

where p is the mean daily percentage of total annual daytime hours for a given time period and latitude. The aBC and bBC coefficients are given by the equations (Allen and Pruitt, 1986; Frevert et al., 1983):

41.1NRH 0043.0 ratiomin −−=BCa (10a)

−++−= dayratiomin U6565.0N0705.1RH00409.0819.0BCb

−0.005968RHminNratio − 0.000597RHminUday (10b)

2.6 Mass transfer method

The mass transfer methods (Aerodynamic) are based on Dalton’s law, which describes the turbulent transfer of water vapour from an evaporating surface to the atmosphere (Singh and Xu, 1997). These methods use simple forms and can be successfully applied when radiation or sunshine


duration data are not available. The basic version of this equation is as follows (Bogawski and Bednorz, 2014):

))(cb(a da2mtmtmt eeuE −+= (11)

where amt, bmt and cmt are constants and ea is the saturation vapour pressure at water surface (kPa). Many variations of Eq. (11) have been proposed (Xu and Singh, 2002). Singh and Xu (1997) evaluated and compared 13 mass transfer equations suggesting that all of these gave almost equally good results. The constant values of the mass transfer equation for Lake Vegoritis (Gianniou and Antonopoulos, 2007), which is in the area of the meteorological station, are amt=2.032, bmt = 1 and cmt =0.26.

2.7 Copais method

The Copais method was developed by Alexandris et al. (2006) to estimate daily reference evapotranspiration ETo using data from solar radiation, temperature and relative humidity. The equation coefficients were estimated using bilinear surface regression analysis. The form of the equation is:

ETo = m1 + m2C2 + m3C1 + m4C1C2 (12)

where m1=0.057, m2=0.227, m3=0.643 and m4=0.0124 and

C1 = 0.6416-0.00784RH+0.372Rs-0.00264RsRH (13)

C2 = -0.0033+0.00812T+0.101Rs+0.00584RsT (14)

where RH in %, T in oC and Rs in MJ m-2 d-1.

2.8 Simplified Penman formula

Valiantzas (2006) proposed a simplified formula for the Penman equation to estimate daily ETo, when wind speed data is missing which is as:

( ) ( ))100/RH(1)20T(09.0R/R4.25.9TR047.0 2asso −++−+=ET (15)

where Ra is the extraterrestrial radiation (MJ m-2d-1).

2.9 Simplified Jensen-Haise method

The Jensen-Haise (Rosenberry et al., 2004) formula has the following form

λρ+=

w

smeano

R)08.0T0025.0(ET

(16)

2.10 Multivariable regression method

Since the meteorological parameters are highly correlated with ETo, multi-linear regression analysis can be used to estimate ETo from these variables. A general form of this equation is as


ETo = m1 + m2Tmean+ m3RH + m4Rs+ m5u2 (17)

where m1, m2, m3, m4 and m5 are regression coefficients.

2.11 Artificial neural networks

Artificial neural networks are non-linear models that make use of a structure capable to represent arbitrary complex non-linear processes that correlate the inputs and outputs of any system (Kisi, 2006; Antonopoulos and Antonopoulos, 2017). The basic unit in an ANN is the neuron (node). Neurons are connected to each other by links known as synapses, associated with each synapse there is a weight factor.

The ANN architecture is defined by the way in which the neurons are interconnected. The network is fed with a set of input-output pairs and is trained to reproduce the output. The structure of each ANN is represented as (i, j, k), where i expresses the number of nodes in the input layer, j the nodes in the hidden layer and k the nodes in the output layer. There is a wide variety of algorithms available for training a network and adjusting its weights. In this article, an algorithm of the multi-layer feed forward artificial neural networks and of the back-propagation for optimization was used. The main task in developing an ANN model is to identify the input variables and the optimal network structure in order to produce the desired output accurately.

2.12 Modeling performance criteria

The model’s performance was evaluated using a variety of standard statistical criteria including the correlation coefficient (r), the root mean square error (RMSE) and the coefficient of efficiency (EF):

⎟⎟

⎠

⎞⎜⎜

⎝

⎛−−−−= ∑∑∑

===

N

i

N

i

N

iCCOOCCOO

1

2mi

1

2mi

1mimi )()())((r

(18)

( )∑=

−=N

iOC

N 1

2ii

1RMSE (19)

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= ∑∑

==

N

i

N

iOO

1

2mi

1

2ii )O()C(1EF (20)

where O are the observed values (the reference evapotranspiration), C are the computed values by the other methods and Om and Cm are the mean observed and computed values, respectively. In most cases the coefficient of determination (R2) is the square of r.

2.13 Study area and data

The weather data that was used in this study to estimate monthly ETo, consisted of monthly data of air temperature, solar radiation, wind speed, and humidity for a period of seven years (2009 to 2015) all measured at the meteorological station of Amynteo in West Macedonia of the northern Greece. It is located in the Ptolemais plain which is surrounded by high mountains (Antonopoulos and Gianniou, 2003). This station is located in the center of an area with four natural lakes surrounding it (Vegoritis, Petron, Zazari and Chemaditis), from which Vegoritis Lake is one of the most significant lakes in northern Greece.


Table 1 presents the maximum, minimum, average and standard deviation values of temperature (oC), relative humidity (%), wind speed (m s-1), solar radiation (W m-2) and ETo from P-M method, for each year of the study period of 7 years. The use of monthly data in this study arises from the fact that meteorological stations rarely have complete daily data and that the data time series present serious limitations and gaps. In most stations in Greece daily meteorological data has been monitored only for the last decade.

Table 1. Statistical parameters of available meteorological variables and ETo at Amynteo station.

Tmean (oC) RHmean (%) u2 (m s-1) Rs (W m-2) ETo (mm d-1)

Max 25.0 95.71 5.047 343.968 6.563 Min 1.7 48.45 0.875 60.780 0.398 Mean 12.7 66.99 2.096 192.592 2.824 sd 7.48 9.74 0.783 88.061 1.870

3. RESULTS AND DISCUSSION

3.1 Comparison results of estimated ETo with deterministic methods

Figure 1 shows the comparison of monthly ETo values predicted by the combination based methods of Penman (PEN), Priestley Taylor (PT), Makkink (MAK), and de Bruin-Keijman (dBRU) and the ETo values of the Penman-Monteith method. Figure 2 presents the comparison between two of the temperature-based methods of Hargreaves (HRG) and Blaney-Criddle (BLC) and the Penman-Monteith method.

Figure 1. Scattering diagrams between ETo with P-M method and a) Penman (PEN), b) Priestley and Taylor (PT), c) Makkink (MAK) and d) de Bruin-Keijman (dBRU) methods.

b)y=1,080x- 0,278R²=0,976

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

EToPT,m

md

-1

EToPM,mmday-1

EToPriestleyTaylor(mm/day)

c)y=1,277x- 0,428R²=0,975

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7

EToMAK,m

md

-1

EToPM,mmday-1

EToMakkink(mm/day)

a)y=0,962x- 0,085R²=0,991

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

EToPEN,m

md

-1

EToPM,mmday-1

EToPenman(mm/day)

d)y=1,074x- 0,189R²=0,972

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

ETodB

RU,m

md

-1

EToPM,mmday-1

EToDeBruin(mm/day)


Table 2 presents the statistical properties of the empirical methods. A lower RMSE value or a higher EF value indicates a lower error between the estimated values from the empirical methods and the values of the P-M method. The Penman method correlated very well with the Penman-Monteith method followed by the Priestley-Taylor, Makkink and de Bruin-Keijman methods. From Figure 1 it can be observed that there is a good correlation between ETo of the Penman-Monteith method and each one of the other methods, which are also consistent with the high values of the coefficient of determination that range from 0.972 to 0.991 and the RMSE values that range from 0.265 to 0.343 mm d-1.

From Figure 2 (and Table 2) it can also be observed that the modified Blaney-Criddle method gives better values of r in relation to the other temperature based method of Hargreaves, but the RMSE value is worst. The values of the coefficient of determination are 0.997 and 0.991 and the RMSE are 1.192 and 0.469 mm d-1, respectively.

Table 2. Statistical criteria of comparison results for 7 years data and prediction by the examined empirical equations.

P-M PEN PT MAK dBRU HRG BLC MTR

Mean 2.824 2.631 2.772 3.179 2.844 3.041 3.455 1.865 sd 1.870 1.807 2.045 2.419 2.037 1.954 2.644 1.343 RMSE

0.265 0.304 0.830 0.343 0.469 1.192 1.054

r

0.996 0.988 0.988 0.986 0.991 0.997 0.986 EF

0.978 0.978 0.881 0.971 0.942 0.794 0.376

P-M mPENI mPENII VAL MLR J-H ETo COP

Mean 2.824 2.703 2.730 4.049 2.824 3.176 4.638 sd 1.870 1.988 1.983 2.576 1.854 2.537 1.876 RMSE

0.209 0.215 1.619 0.310 0.958 2.159

r

0.994 0.993 0.993 0.938 0.988 0.887

EF

0.989 0.988 0.600 0.975 0.856 -0.341

Figure 3 shows the scattering diagrams between monthly ETo values predicted by the Penman-

Monteith method and the modified Penman (mPENI, mPENII, when c is estimated according to Frevert et al. (1983) or Kotsopoulos and Babajimopoulos (1997), respectively) equations (Eq 3) and the simplified methods of Jensen-Haize and Valiantzas (Eq. 10 and 12). From this Figure and the statistical criteria of Table 2, the Penman and modified Penman equations give similar results, while the simplified equations overestimate the ETo values with an r of 0.993 and 0.988 and an RMSE of 1.619 and 0.958 mm d-1, respectively for Valiantzas and Jensen-Haize equations.

Figure 2. Scattering diagrams between ETo with P-M method and a) Blaney-Criddle (BLC) and b) Hargreaves (HRG) methods.

Figure 4 shows the scattering diagrams between monthly ETo values predicted by the Penman-Monteith method and Copais (COP) and multi-linear regression (MLR) equations (Eq. 13 and 17).

y=1,035x+0,116R²=0,981

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

EToHRG

EToPM,mmday-1

EToHargreaves

y=1,409x- 0,525R²=0,994

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8

EToBLC

EToPM,mmday-1

EToBlaney-Criddle


From this Figure and the statistical criteria of Table 2 the MLR equation gives better results compared to the COP equation with r of 0.938 and 0.887 and RMSE of 0.310 and 2.159 mm d-1, respectively.

Figure 5 presents the differences between each one of the ETo estimation methods and the P-M method using the average differences for each month of the time period from 2009 to 2015. The radiation based method of Makkink overestimates ETo values. The other methods of this group present difference of less than a 1 mm d-1, which are observed during the summer months of high evapotranspiration conditions. From the temperature based methods, Blaney-Criddle overestimates the ETo values with differences that approach the 2 mm d-1, while Hargreaves method presents smaller differences. The modified formulas of Penman methods present smaller differences, while the Jensen-Haizen method overestimates the ETo values with significant differences. The methods of Copais, Valiantzas and mass transfer present significant differences to P-M method. Coppais and Valiantzas overestimate the ETo values, while the mass transfer underestimates them.

Figure 3. Scattering diagrams between ETo with P-M method and a) Modified Penman (mPENI), b) Modified Penman (mPENII), c) Jensen-Haise (JH) and d) Valiantzas (VAL) methods.

Figure 4. Scattering diagrams between ETo with P-M method and a) Copais (COP) and b) multi-linear regression (MLR) methods.

c)y=1,340x- 0,609R²=0,976

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

EToJH

EToPM,mmday-1

Jensen-Haise

d)y=1,368x+0,186R²=0,986

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9

EToVA

L

EToPM,mmday-1

Valiantzas

a)y=1,071x- 0,274R²=0,983

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

ETomPENI

EToPM,mmday-1

EToModifiedPenman(F)

b)y=1,053x- 0,244R²=0,986

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

ETomPENII

EToPM,mmday-1

EToModifiedPenman(K-B)

Γραμμική(EToModifiedPenman(K-B))

a)y=0,889x+2,126R²=0,785

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

EToCO

P,m

md

-1

EToPM,mmday-1

Copais

b)y=0,983x+0,046R²=0,983

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

EToMLR,m

md

-1

EToPM,mmday-1

EToMultiRegression


3.2 Comparison results of ETo estimated with ANNs

The architecture of the ANN model was identified by the trial and error procedure (Antonopoulos and Antonopoulos, 2017). The most appropriate one that was finally chosen was the 4-6-1 structure, with 4 neurons in the input layer, 6 neurons in the hidden layer and 1 neuron in the output layer which corresponds to the reference evapotranspiration, using a sigmoid transfer function. Before training and testing, the 4 variables (T, RH, Rs, u2) were standardized and were used as input variables. The target output variable ETo was also standardized before training and testing. All of the available data sets were used for training and testing the model. Detailed information on the data that was split for training and testing, the selection of ANN’s architecture etc, was presented in Antonopoulos and Antonopoulos (2017) in which daily meteorological and ETo data time series was used.

Figure 6 shows the comparison between monthly ETo values of ANNs models of a) 4-6-1, b) 3-6-1, c) 2-6-1 and d) 1-6-1 structure and the ETo values of Penman-Monteith method. Table 3 gives the values of RMSE and r for each combination of the number of input variables. The values of these criteria, when daily data sets were used (Antonopoulos and Antonopoulos (2017), are also presented for comparison reasons. From Figure 6 and Table 3, it can be observed that the ANNs models describe the monthly values of ETo with high accuracy, which however decreases as the number of input variables lessens. The RMSE values of all combinations in the input variables ranged from 0.509 to 0.947 mm d-1 and the r values from 0.955 to 0.998. This shows that using even 3 or 2 input variables in the ANN models also gives high accuracy results in the ETo evaluation.

Table 3. Statistical criteria of results obtained using different number of input variables of ANN models. The data sets

of 2009-2015 were used for training and testing.

ETo PM 4 Input variables Τ, RH, Rs, u2

3 Input variables Τ, RH, Rs

2 Input variables Τ, Rs

1 Input variable Τ

Mean (mm d-1) 2.821 2.821 2.821 2.815 2.821 sd 2.259 2.247 2.235 2.229 2.167 RMSE 0.509 0.638 0.663 0.947 r 0.998 0.992 0.99 0.955

For daily datasets RMSE 0.574-1.326 0.598-0.954 0.640-0.871 1.020 r 0.955-0.986 0.952-0.978 0.951-0.976 0.897

3.3 Methods’ adjustment to local conditions

Many of the ETo methods presented in this paper contain coefficients that were estimated for specific locations. The Pristley-Taylor, Makkink, de Bruin-Keijman, Valiantzas, and Blaney-Criddle methods were modified by adjusting coefficients to obtain better comparison with ETo P-M values. The software of Wessa (2017) was used in these estimations. The modified equations are presented in Table 4, with the values of statistical criteria. The PT and Makkink methods give results nearly identical to the original form of the equations. The adjusting mass transfer and Valiantzas equations have been improved significantly in relation to the original form of the equations. From the multilinear regression equations, these which involve the three basic variables (Rs, Tm, RH) present the better results of statistical criteria. These values compared well with the values of the Penman and PT methods. The equation with Rs and T also show good statistical values, while the equation with only T gives statistical values that are lower but remain still high. Using Ra as radiation in the multilinear equation with 3 or 2 variables, results in high values of statistical criteria, almost identical to the case of using Rs.


Figure 5. Mean monthly differences of ETo of different method estimation from P-M method for period of 2009 to 2015.

4. DISCUSSION

Many researchers worked with different methods of evapotranspiration from land covered surfaces or evaporation from free surface water of lakes based on the number of variables that are affecting them. The methods may be categorized into one of the following classes of energy budget, aerodynamic transfer (or mass transfer), a combination of these and simplified empirical methods or multilinear regression methods.

A performance rank from the best to the least effective is determined by the RMSE and EF values and it is given in Table 5. From this table the rank is: mPENI, mPENII, PEN, PT, MLR, dBRU, HRG, MAK, J-H, MTR, BLC, VAL and COP. From the radiation-temperature based category, Penman and the modified Penman, Priestley Taylor and de Bruin-Keijman are among the best methods. From the temperature based category, Hargreaves is better compared to Blaney-Criddle.

Evaporation is one of the most important components in both the energy and water budgets of lakes and a primary process of water loss from their surface (Antonopoulos et al., 2016). Accurate estimation of lake evaporation is necessary for water and energy budget studies, lake level forecasts, water quality surveys, water management and planning for hydraulic constructions. Evaporation methods that include available energy and aerodynamic terms have been studied in comparison to the Bowen-ration. Energy-Budget was used as the standard method in many articles (Rosenberry et al., 2007; Winter et al., 1995; Dalton et al. 2004; Antonopoulos et al. 2016). They found that empirical methods that emphasize on the assessment of energy fluxes provide the best estimates of water loss to the atmosphere. Rankings of alternate methods for estimating evaporation varied among the three above-mentioned studies. In some cases, the robustness of a particular empirical method depended on the ambient climate.

-1,0

0,0

1,0

2,0

ΔΕΤo,m

md-

1

PM-PEN PM-PT PM-MAK PM-HRG

-1,0

1,0

3,0ΔΕΤo

,mmd

-1PM-HRG PM-BLC

-1,0

0,0

1,0

2,0

ΔΕΤo,m

md

-1

PM-mPENII PM-mPENI PM-JH

-3,0

-1,0

1,0

3,0

ΔΕΤo,m

md

-1

NoofmonthPM-COP PM-MLR PM-VAL PM-MTR


Figure 6. Scattering diagrams between ANNs results and P-M method of ETo (mm d-1) during the testing periods, when monthly data of 2009-15 were used for training and testing the ANN model and a) T, RH, u2 and Rs b) T, RH, Rs, c) T,

Rs and d) only T are the input variables, relatively.

Table 4. Equations adjustment to local conditions and statistical criteria.

Mean sd RMSE r EF

Priestley-Taylor λρ

−+

=w

G232.

)( nR1ETγΔΔ

o (22)

2.711 2.0 0.304 0.988 0.988

Makkink λρΔ+=γ+Δ w

so

R465.0304.0ET

(23)

2.818 1.844 0.830 0.988 0.988

Mass transfer

))(092.01(6073.3 ds2 eeuETo −+= (24) 2.824 1.813 1.054 0.986 0.986

Valiantzas ( ) +−+= 2

asso R/R1136.25.9TR0303.0ET

( ) (25) )100/RH(1)20T(10168.0 −++

2.824 1.858 1.619 0.993 0.994

Blaney-Criddle

))13.8T46.0(p(171.1827.1 ++−=oET (26) 2.822 1.838 0.302 0.983 0.973

MLR-3v T097964.0R072025.07904.2 a ++=oET RH047816.0− (27) 2.824 1.839 0.362 0.983 0.961

MLR-2v T112958.0R0971096.029661.1 a ++−=oET (28) 2.824 1.816 0.387 0.971 0.954

MLR-1v T236434.0184415.0 +−=oET (29) 2.824 1.769 0.606 0.946 0.881

MLR-4v ++−+−= 2u24879.0RH 014304.0T1001.025616.0oET 0.013654 Rs (30) 2.824 1.855 0.310 0.992 0.972

MLR-3v −++= T096501.0R12885.005127.1 soET 0.023483RH (31) 2.824 1.845 0.319 0.987 0.970

MLR-2v T092842.0R158265.0958718.0 s ++−=oET (32) 2.824 1.841 0.294 0.985 0.974

The evaluation of empirical equations of reference evapotranspiration in relation to the Penman-

Monteith method (standard method) has been also the subject of many articles. The main reasons of these studies are 1) the poor data availability that limits the use of the P-M method in many regions and 2) the need to calibrate the empirical equations under the regional conditions. Hargreaves method, as an example, is one of the most used methods, because it requires limited variables but it

d) y = 0,908x + 0,255R² = 0,912

01234567

0 1 2 3 4 5 6 7

ET

o A

NN

1v,

mm

d-1

ETo PM, mm d-1

c) y = 0,975x + 0,063R² = 0,979

01234567

0 1 2 3 4 5 6 7

ET

o A

NN

2v,

mm

d-1

ETo PM, mm d-1

b) y = 0,981x + 0,051R² = 0,983

01234567

0 1 2 3 4 5 6 7

ET

o A

NN

3v,

mm

d-1

ETo PM, mm d-1

a) y = 0,993x + 0,016R² = 0,996

01234567

0 1 2 3 4 5 6 7

ET

o A

NN

4v,

mm

d-1

ETo PM, mm d-1


overestimates ETo at wet conditions and underestimates it at more dry conditions and high wind velocities (Allen et al., 1998; Garcia et al., 2004).

The radiation and temperature based methods generally describe the monthly ETo in the examined area of northern Greece better. The temperature based methods follow in accuracy and the mass transfer and empirical methods with less accuracy. The adjustments of coefficients of some empirical methods to the local conditions significantly improve the results of ETo estimation. Multilinear regression equations produce accurate results even when less meteorological variables are used in the equation. These results and conclusions are similar to the results of many other authors that examined empirical equation of ETo, as they are presented below. Xu and Singh (2002) evaluated and compared five empirical reference evapotranspiration equations from the three categories of temperature-based (Hargreaves and Blaney-Criddle), radiation-based (Makkink and Priestley-Taylor) and mass-transfer-based methods with the Penman-Monteith equation using daily meteorological data from the Changins station in Switzerland. The study showed that: (1) the original constant values involved in each empirical equation worked quite well for the study region, except that the value of α = 1.26 in Priestley-Taylor was found to be too high and the recalibration gave a value of α = 0.90 for the region, (2) improvement was achieved for the Blaney-Criddle method by adding a transition period in determining the parameter p, (3) the differences of performance between the best equation forms selected from each category are smaller than the differences between different equations within each category as reported in earlier studies (Xu and Singh, 2000). Further examination of the performance resulted in the following rank of accuracy as compared to the Penman-Monteith estimates: Priestley-Taylor and Makkink (radiation-based), Hargreaves and Blaney-Criddle (temperature-based) and Rohwer (mass-transfer).

Lu et al. (2005) studied six commonly used ETo methods using datasets from forested watersheds in the southeastern United States. The temperature based Hargreaves-Samani and the radiation based Makkink, and Priestley-Taylor ETo methods were compared. The study found that ETo values calculated from the six methods were highly correlated (Pearson Correlation Coefficient 0.85 to 1.00). The Priestley-Taylor method performed better than the other ETo methods.

Table 5. Ranked performance of examined empirical method according to RMSE and EF values.

Method RMSE, mm d-1 EF Rank mPENI 0.209 0.989 1 mPENII 0.215 0.988 2 PEN 0.265 0.978 3 PT 0.304 0.978 4 MLR 0.310 0.975 5 dBRU 0.343 0.971 6 HRG 0.469 0.942 7 MAK 0.830 0.881 8 J-H 0.958 0.856 9 MTR 1.054 0.376 10 BLC 1.192 0.794 11 VAL 1.619 0.600 12 COP 2.159 -0.341 13

Xystrakis and Matzarakis (2011), using daily meteorological data from seven meteorological

stations on the island of Crete (southern Greece), evaluated 13 empirical equations (radiation- and temperature-based) of ETo. The radiation-based equations generally exhibited better fits than the temperature-based equations. Jensen-Haise equation, Makkink equation and De Bruin equation clearly revealed a great bias and, therefore, their use for the estimation of ETref in the semiarid climate of Crete is not recommended.

Rácz et al. (2013) presented a comparative analysis of different methods of ETo evaluation for the continental climate of Eastern Hungary. They concluded that Priestley-Taylor, Penman−Monteith−FAO-56, parameterized with alternative radiation balance and Makkink methods were found to be the best performing models for ET calculation. According to Efthimiou et al. (2013), the Pristley-Taylor method had the best correlation to the ETo-PM, amongst many other


empirical equations as Penman, Makkink, Hargreaves and Copais, while using the data from two meteorological stations at Western Macedonia, Greece.

Djaman et al. (2015) evaluated the performance of 16 ETo equations against the ASCE-PM equation under the sahelian conditions in the Senegal River Valley. The Hargreaves and the modified Hargreaves equation systematically overestimated ETo. The equations of Makkink–Hansen systematically underestimated ETo. Temperature-based equations performed relatively better. Mass transfer equations of Trabert and Mahringer also had better performance and represent simple ETo estimation methods that could be used under conditions of limited climate data in the Senegal River Valley. The results of Heydari et al., (2014) from a comparison and evaluation study of 38 equations to estimate ETo in an arid region, showed that the mass transfer-based equations had the worst performances among the equations evaluated. The ETo values obtained from the equations, derived specifically for this location, were better than those estimated by existing equations.

Recently a significant number of articles use artificial intelligence techniques to simulate evaporation from free water surface as well as actual and reference evapotranspiration (Kumar et al., 2011). The results of ANN models in evaluation of ETo with the available datasets of our study area from the region of West Macedonia in Northern Greece showed that they have the ability to describe it with high accuracy. This accuracy is in agreement with the results of radiation and temperature based methods of Penman, modified Penman, Pristley-Taylor, de Bruin-Keijman and those of the empirical methods. Similar are the results of using ANNs models from other authors as they are presented below. The results of Terzi and Keskin (2010) evaluation of artificial neural networks (ANN) models and some classical methods showed that PT and MAK methods underestimated evaporation values and that the ANN model is superior even to classical methods. Diamantopoulou et al. (2011) compared two ANN models and HRG method to the P-M method at three meteorological stations in the central and northern Greece. The estimated r and RMSE values of the HRG method ranged from 0.914 to 0.958 and 0.553 to 0.725 mm d-1, respectively, while those for the ANN models ranged from 0.934 to 0.957 and 0.398 to 0.444 mm d-1, respectively. Traore et al. (2010) also showed that the ANNs temperature-based model has a better performance when compared to the empirical Hargreaves method. Gocic et al. (2016) analyzing the ETo using an extreme learning machine (ELM) and the adjusted Hargreaves, Priestley–Taylor and Turk, obtained results which revealed that the ELM and Hargreaves method was superior to the other conventional methods. Shiri et al. (2014) applied ANNs and some other Heuristic Data Driven (HDD) models to study ETo and to compare it with the corresponding Hargreaves, Makkink, Priestley–Taylor, Turc methods, and non-linear calibrated methods. The calibrated Hargreaves model was found to give the most accurate results in all local and pooled scenarios, followed by the calibrated Priestley–Taylor model. The comparison results of Antonopoulos and Antonopoulos (2016) of ANNs models and empirical equations showed that ANNs models estimate ETo with an accuracy of a RMSE that ranged from 0.574 to 1.33 mm d-1, and r from 0.955 to 0.986. The Priestley-Taylor and Makkink methods correlated very well (RMSE of 0.962 and r of 0.925) with the Penman-Monteith method followed by the Hargreaves method. The mass transfer method also correlated sufficiently (RMSE=0.922, r=0.849) but it underestimated the ETo values.

As Jain et al. (2008) noted, the ANNs have the ability to learn from the data (training datasets) to recognize a pattern in the data, to adapt solutions and to process information rapidly. The major disadvantage of ANNs in hydrological models is that they do not explain the underlying physical processes in mapping the relationships while operating as “black box” models. Another disadvantage is that a model developed for one specific location cannot be implemented to other locations without local training. The most serious disadvantage is that these models require significant effort in order to be used, which makes them very difficult for a common user without any software (as MATLAB, or STATGRAPHICS) and structure knowledge of these models. In contrast the empirical models are very simple for anyone to use and require no special knowledge and skill.


5. CONCLUSIONS

Monthly reference evapotranspiration rates were estimated with the combination type method of Penman – Monteith. The results of alternative empirical methods of radiation-temperature, temperature, mass transfer based methods and some others equations based on regression or simplified approaches were compared with the ETo-PM. ANN models were also used to describe ETo and to compare it with the empirical methods. Seven years (2009 to 2015) of monthly data from the station of Amynteo near the Lake Vegoritis in the northern Greece were used. The Penman Monteith method is considered the standard method to compare and evaluate the empirical/deterministic models of reference evapotranspiration.

The empirical temperature and radiation based methods gave the best results followed by the temperature based methods, while the simplified or regression equations had less accuracy. The multilinear regression equations, based on the local meteorological datasets, gave accurate results either by using all input variables or by limiting them. The recalibration of the Pristley-Taylor, Makkink, Blaney-Criddle, Valiantzas and mass transfer methods with the local meteorological datasets improved the results of the ETo estimation.

The accuracy and statistical criteria of the (4-6-1) structure ANN model is considered the best of all the other ANN structures. The performance of the ANNs models while using less input variables, but containing temperature and solar radiation, was also adequate and gave good results with high accuracy. Generally, the accuracy of the ANN models is similar to the one of the radiation and temperature based empirical methods.

The empirical methods to estimate reference evapotranspiration still remain a valuable tool, as it is one of the main components of the hydrological balance, and because of the high interest in the hydrological and agronomical studies. Its accurate estimation over the world, under local conditions, is also very important for the water resources management. The newest methods of artificial intelligent processes, while giving accurate results, require special knowledge from the users and significant dataset of the same variables that empirical methods use to estimate ETo.

ACKNOWLEDGEMENTS

The author acknowledges Municipality of Amyntaio in Greece for providing the data sets from the meteorological station.

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