Theor Appl Climatol DOI 10.1007/s00704-016-1744-7
ORIGINAL PAPER
Calibration of Valiantzas’ reference evapotranspiration equations for the Pilbara region, Western Australia Matin Ahooghalandari 1 & Mehdi Khiadani 1 & Mina Esmi Jahromi 1
Received: 18 May 2015 / Accepted: 24 January 2016 # Springer-Verlag Wien 2016
Abstract Reference evapotranspiration (ET0) is a critical component of water resources management and planning. Different methods have been developed to estimate ET0 with various required data. In this study, Hargreaves, Turc, Oudin, Copais, Abtew methods and three forms of Valiantzas’ formulas, developed in recent years, were used to estimate ET0 for the Pilbara region of Western Australia. The estimated ET0 values from these methods were compared with those from the FAO-56 Penman-Monteith (PM) method. The results showed that the Copais methods and two of Valiantzas’ equations, in their original forms, are suitable for estimating ET0 for the study area. A modification of Honey-Bee Mating Optimization (MHBMO) algorithm was further implemented, and three Valiantzas’ equations for a region located in the southern hemisphere were calibrated.
1 Introduction Water resources management is a worldwide challenge. In Australia, water resources are characterized by a high level of usage and annual and seasonal variability in precipitation, soil moisture, and the subsequent groundwater recharge. For * Mehdi Khiadani
[email protected] Matin Ahooghalandari
[email protected] Mina Esmi Jahromi
[email protected]
1
School of Engineering, Edith Cowan University, Joondalup, WA 6027, Australia
example, one of the worst droughts in recorded history occurred in Australia earlier this century (Alexander et al. 2007, Liu et al. 2007). Efficient and sustainable management of water resources requires precise knowledge of the generation and distribution of the resources. In a water balance system, along with precipitation, surface and groundwater flow, evapotranspiration (ET) is a key component (Du et al. 2013). This consists of two processes: (1) direct evaporation from open water bodies, vegetation, and bare ground and (2) transpiration that comprises the transport of water from the soil/ aquifer system through plant roots and steam from the plant leaves into the atmosphere (Sen ay et al. 2011). Evapotranspiration can be measured using micrometeorological techniques (e.g., lysimeters), catchment water balance techniques (Ford et al. 2007, Wilson et al. 2001), and using empirical and/or physically-based equations (Alexandris et al. 2006). Direct measurement gives more accurate values than empirical methods; however, that is expensive and requires complex instrumentation (Vaughan and Ayars 2009). So, evapotranspiration (ET 0 ) estimation through empirical methods is an appropriate option. In recent decades, the applicability of different methods has been examined for estimating ET0 (Landeras et al. 2008, Sabziparvar et al. 2010, Tabari et al. 2013; Trajkovic and Kolakovic 2009a, Valiantzas 2012a, Valiantzas 2012b, Valiantzas 2012c). For estimating evapotranspiration, the FAO-56 Penman-Monteith (PM) method was ranked as the best method for precisely estimating ET0 relative to lysimeter results in all climate conditions (Beyazgül et al. 2000, Hussein 1999, Kisi 2013, Trajkovic 2005, Tyagi et al. 2003). The International Commission for Irrigation and Drainage and Food and Agriculture Organization of the United Nations (FAO) also proposed using the PenmanMonteith equation as a standard method for estimating reference evapotranspiration (Trajkovic 2005). This method (Allen et al. 1998) uses a hypothetical green grass with an assumed
Ahooghalandari M. et al.
height of 0.12 m, a surface resistance of 70 s m−1, and an albedo of 0.23 (Jabloun and Sahli 2008). The FAO-56 PM estimated ET0 as follows:
ETo ¼
900 u 2 ð es ea Þ T mean þ 273 Δ þ γ ð1 þ 0:34u2 Þ
0:408ΔðRn GÞ þ γ
ð1Þ
where ET0 = reference evapotranspiration (mm day−1), G = soil heat flux density (MJ m−2 day−1), Rn = net radiation (MJ m−2 day−1), γ = psychometric constant (KPa0C−1), Tmean = mean daily air temperature (0C), u2 = average 24-h wind speed at 2 m height (m s−1), Δ = slope of the saturation vapor pressure function (KPa0C−1), and es − ea. = vapor pressure deficit (kPa). The main barrier in using the FAO-56 PM method is the number of required data which may not be available at many weather stations. The quality of data is also a serious problem (Jensen et al. 1997, Trajkovic 2005) Several ET0 estimation methods have been developed (Hargreaves et al. 1985; Jensen and Haise 1963; Makkink 1957; Priestley and Taylor 1972; Thornthwaite 1948, Turc 1961, Valiantzas 2006). All these methods are purely empirical methods. On the contrary, Valiantzas semi-empirical equations are obtained from various simplifications of Penman equation. These methods vary in terms of complexity and required data. However, selecting an appropriate method for estimating ET 0 depends on the location, climate conditions, intended estimation period, and the availability of input data (Abdel-Wahed and Snyder 2008, Azhar and Perera 2010; Trajkovic and Kolakovic 2009a) DehghaniSanij et al. (2004) compared ET 0 estimates, from six different methods including Penman, Penman-Monteith, PenmanWr i g h t , B l a n e y – C r i d d l e , r a d i a t i o n b a l a n c e , a n d Hargreaves, with experimentally determined ET0 values in a semi-arid environment in Iran; the Penman-Monteith method generated the most consistent estimates of ET 0 compared to lysimeter data. Jun et al. (2008) compared the outputs from ten different methods of estimating ET0 versus the PM method in the Loess Plateau of China; a good correlation between PM and Kimberly Penman methods was noticed. López-Urrea et al. (2006) applied seven empirical methods for estimating ET 0 in the province of Albacete in Spain with a semi-arid climate and indicated that the Hargreaves and Samani (1985) method proving to be the most accurate. Five methods of estimating ET0 including Hargreaves and Samani (1985); Thornthwaite (1948); Turc (1961); Priestley and Taylor (1972), and Jensen and Haise (1963) were evaluated by Trajkovic and Kolakovic (2009a) using the climatic data from seven humid weather stations in the Western Balkans in Europe (Varazdin, Palic, Zagreb-Maksimir, Belje, Novi Sad, Negotin, and Nis). The Turc equation was the most suitable
method. George et al. (2002) evaluated eight methods of estimating ET0 at three sites in India and California. They showed that the Turc (1961), and Priestley and Taylor (1972) methods underestimate ET0 in humid climate conditions. Alexandris et al. (2006) developed the BCopais^ empirical function for estimating daily reference evapotranspiration using solar radiation, temperature, and relative humidity, and the results were well compared with ASCE Penman-Monteith, CIMIS–Penman, FAO-56 PM, and daily Hargreaves–Samani methods. Meyer (1999) argued that no single method of estimating ET0 without some local calibration is universally satisfactory for all climatic conditions; and it may be essential to calibrate a method versus properly measured ET0 (Trajkovic and Kolakovic 2009b). Trajkovic (2007) calibrated the Hargreaves equation for estimating ET0 in the Western Balkans, and contrary to 0.5 in the original equation, a value of 0.424 was recommended. As an alternative version of the Thornthwaite (1948) scheme, Pereira and Pruitt (2004) proposed a photoperiodic effective daily temperature for estimating daily ET0. Zhai et al. (2010) examined eleven empirical evapotranspiration equations including Hargreaves, Makkink, Turc, Priestley–Taylor, Jensen–Haise, Doorenbos and Pruitt, Abtew, McGuinness and Bordne, Rohwer and Blaney– Criddle in Gansu Province in China and the results well compared with measured pan evaporation data. Fooladmand and Haghighat (2007) calibrated the Hargreaves equation against the Penman-Monteith method for 14 synoptic stations in Iran, and recommended monthly coefficients for estimating ET0. Based on the previous literature, there is no clear agreement about the most appropriate method of ET 0 estimation. Therefore, estimation of ET0 from a distinct method depends on the location and climate, and if the validation process fails, calibration may improve the results. The objectives of this study are (1) to evaluate performance of different methods of estimating reference evapotranspiration for the Pilbara region of Western Australia and (2) to calibrate Valiantzas’ evapotranspiration equations versus the FAO-56 PM method using a modification of Honey-Bee Mating Optimization (MHBMO) algorithm.
2 Material and methods 2.1 The study area The Pilbara region of Western Australia extends from the Indian Ocean to the west, and the Northern Territory border to the east (Longitude 129° E), covering more than 500,000 km2 (Fig. 1). The Pilbara’s climate is classified as semi-arid with two distinct seasons of hot and
Calibration of Valiantzas’ reference evapotranspiration equations Fig. 1 The study area
wet summers (October to April), and mild winters (May to September). Rainfall in the Pilbara is influenced by two primary climatic systems, a northern rainfall system associated with tropical lows and a winter rainfall event associated with low pressure frontal systems with an average annual rainfall of 330 mm that mostly falls in summer, during cyclonic activities, or thunderstorms. There are enormous changes in temperature from summer to winter; it exceeds 40 °C in summer, while varies between 14 and 28 °C in winter (Van Vreeswyk 2004). 2.2 Methodology In this study, ten methods of estimating ET0 including Hargreaves, Turc, Oudin, Copias, Abtew, and three forms of Valiantzas’ equations were used for the Pilbara region. They have been given in Table 1 and were selected based on their popularity, required input data, and the available data. Daily time step for the input parameters for each method and the estimated ET0 from the FAO-56 PM were obtained from the Australian Bureau of
Meteorology (BOM) and the Scientific Information for Land Owners (SILO), respectively. Table 2 presents the weather station information including names, latitude and longitude coordinates, elevation, and weather parameters. Unfortunately, no field measurement of ET0 was available for the study area to be used for validation of the methods used. Therefore, the ET0 values estimated from each method was assessed versus the estimated values from the FAO-56 PM provided by SILO. The Valiantzas equations were developed based on the climatic data from the northern hemisphere and correlated well with the calculated values from the FAO-56 PM. Valiantzas suggested different equations for estimating ET0 values which depend on the availability of climatic data at a specific location (Valiantzas 2012a, Valiantzas 2012c). The equations (Valiantzas 2006, Valiantzas 2012a, Valiantzas 2012c) were originally derived from the Penman equation and verified using the climatic observations in the northern hemisphere; as such, they have not been examined for the southern hemisphere climate conditions.
Ahooghalandari M. et al. Reference evapotranspiration estimation methods
Table 1 Method
Reference
Representative equation
Hargreaves (RS,T) Turc (RS,T,RH)
(Hargreaves 1975)
*
(Turc 1961)
**
ETO = 0.0135Rs(T + 17.8)
ETO ¼ 0:013
T T þ15
RH < 50% ðRs þ 50Þ 1 þ 50RH 70
T ETO ¼ 0:013 T þ15 ðRs þ 50Þ RH ≥50% * þ5 ETo ¼ Ra T100 **
Oudin (T)
(Oudin et al. 2005)
Copais (RS,T,RH)
(Alexandris et al. 2006)
Abtew (RS,Tmax) Valiantzas 1 (RS,T,RH) Valiantzas 2 (RS,T,RH) Valiantzas 3 (RS,T,Tmin)
(Abtew 1996) (Valiantzas 2012a, 2012c)
***
ETo = m1 + m2C2 + m3C1+m4C1C2 C1 = 0.6416 - 0.00784RH + 0.372Rs - 0.00264RsRH C2 = - 0.0033 + 0.00812T + 0.101Rs + 0.00584RsT m1 = 0.057 m2 = 0.227 m3 = 0.643 m4 = 0.0124 * ETo = 0.01786RsTmax 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *** RH RH þ 0:1W aero ðT þ 20Þ 1 100 ETo ¼ 0:0393Rs T þ 9:5 2:4 RRas 0:024ðT þ 20Þ 1 100 RH > 65 % Waero = 0.78 RH ≤ 65 % Waero = 1.067 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *** RH ETo ¼ 0:0393Rs T þ 9:5 0:19Rs0:6 φ0:15 þ 0:078ðT þ 20Þ 1 100 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi *** ETo ¼ 0:0393Rs T þ 9:5 0:19Rs0:6 φ0:15 þ 0:0061ðT þ 20Þð1:12T T min 2Þ0:7
Rs global solar radiation, Ra extraterrestrial radiation, RH relative humidity (%), T average temperature (o C), Tmax maximum temperature (o C), Tmin minimum temperature (o C), Waero wind function, φ latitude (rad) 1
*Ra and Rs in mm/d; **Rs in cal cmd ; ***Ra and Rs in MJm-2 /d
For each method in Table 1, the statistical parameters of mean bias error (MBE), root mean square error (RMSE), relative error (RE), and the coefficient of determination (R2) were estimated from (Jacovides and Kontoyiannis 1995): 1 ∑ ðET0est ET0ref Þ mm day1 ð2Þ n i¼1 0:5 1 n 2 ∑ ðET0est ET0ref Þ mm day1 ð3Þ RMSE ¼ n i¼1
MBE ¼
RE ¼
Table 2
2 ∑ ET0est ET0est ETiref ETiref i¼1 R2 ¼ n 2 n 2 ð5Þ ∑ ETiest ETiest ∑ ETiref ETiref n
i¼1
i¼1
n
RMSE EToref
100
ð4Þ
where ET0 - est is the estimated value of ET0, ET0 - ref is the reference ET from the FAO-56 PM, and n is the number of data. The methods with lower MBE and RMSE, and higher R2 were considered to have excelling performance among others.
Weather stations details located in the Pilbara region
Station number
Station name
Latitude (decimal)
Longitude (decimal)
Station height (mAHD)a
Average rainfall (mm)
Average temperature (°C)
Relative humidity (%)
Wind speed (m/s)
1 2 3 4 5 6 7 8
Wittenoom Marble Bar Pannawonica Gascoyne Junction Meekatharra Airport Onslow Airport Port Headland Airport Telfer Aero
−22.2425 −21.1756 −21.6392 −25.0544 −26.6136 −21.6689 −20.3725 −21.7125
118.3358 119.7497 116.3308 115.2100 118.5372 115.1092 118.6317 122.2281
463.00 182.30 200.00 144.00 517.00 10.50 6.40 291.90
463 371 405 212 237 319 318 369
26.7 27.8 27.1 24.4 22.5 25.5 26.3 26.7
19–64 20–70 25–72 26–70 18–67 41–74 23–75 17–65
0.6–5 0.7–5 1–2.8 0.7–11 2.7–4.8 2.4–5.2 1.9–6.3 3.3–5.7
a
Australian height datum
Calibration of Valiantzas’ reference evapotranspiration equations
2.3 Procedure adopted for calibration An artificial intelligent model based on the Honey-Bee Mating Optimization algorithm (HBMO) was used to calibrate and optimize the three equations were recently suggested by Valiantzas for estimating ET0. The HBMO algorithm stimulates the social behavior of a bee colony (Esmi Jahromi and Afzali 2014) consisting of a single egg laying queen (female), drones (males), and workers and broods (Moritz and Southwick 1992). These four groups perform special tasks in their society so that the general condition of the colony improves. The HBMO algorithm basically simulates the natural mating process of the queen when she leaves the nest to mate with drones (Sabar et al. 2012). It has been applied to various optimization problems (Yuce et al. 2013). In the mating process, the drones mate with the queen probabilistically, using an annealing function as (Abbass 2001): Δð f Þ
ProbðQ; DÞ ¼ exp SðtÞ
ð6Þ
where Prob (Q, D) is the probability of adding the sperm of drone D to the spermatheca of queen Q, Δ( f ) is the absolute difference between the fitness of
drone and the fitness of queen, and S(t) is the speed of the queen at time t (Marinakis et al. 2011). At the beginning of mating flight, the queen’s speed and energy are high and then decrease after each transition in space as follows: Speedðt þ 1Þ ¼ α Speedðt Þ
ð7Þ
Energyðt þ 1Þ ¼ Energyðt Þ γ
ð8Þ
where α is a factor in the range of [0,1] and γ is the amount of energy reduction after each transaction. The drone’s sperm is stored in the queen’s spermatheca if the mating is successful (i.e., the drone passes the probabilistic decision rule) (Haddad et al. 2006). To improve the brood’s genotype, the workers use a crossover mechanism. The quality of a new brood is fittest due to different solution parts, and this fitness is determined by evaluating the value of objective function of the brood genotype (Esmi Jahromi and Afzali 2014). In the HBMO, the proposed algorithm of brood generation often converges to a local optimum which is a disadvantage of this method. In order to avoid this, the Modified Honey-Bee Mating Optimization (MHBMO) algorithm, as shown in Fig. 2, has been suggested for improving the
Fig. 2 Flowchart of the MHBMO algorithm
Start
Input data Keep the previous queen
Initial population generation
No
Is the best solution better than the queen?
Yes
Substitute the best solution with the queen
Generate queen spermatheca matrix
Sorting based on RSME
K=1
Queen selection
No
Results Yes
Select three sperms randomly
Are the termination criteria satisfied?
Generate three broods including the original HBMO brood and two improved broods
Select the best solution
Calculate RMSE for the generated broods
Sorting based on RMSE
Calculate objective functions for the new colony
K=K+1
Consider the best brood as the k-th brood with minimum RMSE among the three generated broods
Feeding generated broods and queen with royal jelly by workers
K 50 %, this equation excludes relative humidity which imply RH has less influence on the calculation of ET0. An investigation of the climate data in the study area showed that more than half of data have RH > 50 % in some weather stations, which this may explain why Valiantzas 3 (RS,T,Tmin) has a better performance.
3.2 Calibration of Valiantzas’ equations The results in Fig. 3 show that the use of Valiantzas 1 and 2 (RS,T,RH) in their original form may be improved by calibrating these equations using the Modified Honey-Bee Mating Optimization (MHBMO) algorithm. The MHBMO algorithm has some advantages over other optimization methods including the feasibility of finding a global optimum for several problems, implementation with several optimization problems, and availability for real and binary problems (Yuce et al. 2013). These encouraged the authors to use the MHBMO for calibrating the Valiantzas’ equations. In this study, one of the considerations given for calibrating these equations was that the original forms of the equations remain unchanged; however, a coefficient as multiplying factor could be introduced to each individual term using MHBMO. The optimization procedures discussed earlier were followed, and the calibrated Valiantzas’ equations for the Pilbara are presented in Table 4. The plots of estimated ET 0 from the calibrated Valiantzas’equations versus FAO-56 PM and the corresponding statistical parameters MBE, RMSE, RE, and R2 are presented in Fig. 4 and Table 5, respectively.
Comparison of the statistical parameters in Tables 3 and 5 shows that the calculated ET0 from the calibrated Valiantzas equations has improved for all three equations. Among these, Valiantzas 2 (RS,T,RH) still has a good performance; however, the most considerable improvement belongs to Valiantzas 1 (Rs,T,RH) with a change in relative error, RE, from 11.11 to 6.5 %. The calculated ET0 from Valiantzas 3 (RS,T,Tmin) slightly improved. The results show that the calibrated Valiantzas 1 and 2 (Rs,T,RH) equations with limited number of parameters can be confidently used to estimate ET0 for the study area.
4 Conclusion Various methods of estimating evapotranspiration were evaluated using climate data from the weather stations in the Pilbara region of Western Australia. Among these, the coefficient of determination, R2, for Copais (RS,T,RH), Valiantzas 1 and 2 (RS,T,RH) methods in their original forms are greater than the other methods. The calculated ET0 from Valiantzas’ equations are well correlated with those from the FAO-56 PM method, which are suitable methods when climate data are limited. Calibration of Valiantzas 1, 2 (RS,T,RH), and 3 (RS, T,Tmin) equations using the MHBMO optimization algorithm improved the estimation of ET0 with the coefficient of determination of 0.950, 0.927, and 0.900, respectively. Valiantzas 1 (RS,T,RH), which contains extraterrestrial radiation (Ra) and wind function terms is the most precise method for the Pilbara region after calibration, when limited climate data are available. Further assessments of the Valiantzas’ equations are recommended for other parts of Australia for validating their suitability under different climate conditions. Acknowledgments The authors would like to thank the Bureau of Meteorology for providing the required data for this study.
References Abbass HA (2001) MBO: Marriage in honey bees optimization—a haplometrosis polygynous swarming approach. Evolutionary Computation, 2001 Proceedings of the 2001 Congress on. IEEE 1: 207–214 Abdel-Wahed MH, Snyder RL (2008) Simple equation to estimate reference evapotranspiration from evaporation pans surrounded by fallow soil. Journal of Irrigation and Drainage Engineering 134:425– 429 Abtew W (1996) Evapotranspiration measurements and modelling for three wetland systems in South Florida. J Am Water Resour Assoc 32:465–473 Alexander LV, Hope P, Collins D, Trewin B, Lynch A, Nicholls N (2007) Trends in Australia’s climate means and extremes: a global context. Aust Meteorol Mag 56:1–18
Calibration of Valiantzas’ reference evapotranspiration equations Alexandris S, Kerkides P, Liakatas A (2006) Daily reference evapotranspiration estimates by the BCopais^ approach. Agric Water Manag 82:371-386 Allen RG, Pereira LS, Raes D, Smith M (1998) Crop evapotranspirationguidelines for computing crop water requirements-FAO irrigation and drainage paper 56. FAO, Rome Azhar AH, Perera B (2010) Evaluation of reference evapotranspiration estimation methods under southeast Australian conditions. J Irrig Drain Eng 137:268–279 Beyazgül M, Kayam Y, Engelsman F (2000) Estimation methods for crop water requirements in the Gediz Basin of western Turkey. J Hydro l229:19–26 DehghaniSanij H, Yamamoto T, Rasiah V (2004) Assessment of evapotranspiration estimation models for use in semi-arid environments. Agric Water Manag 64:91–106 Du J, Song K, Wang Z, Zhang, B, Liu D (2013) Evapotranspiration estimation based on MODIS products and surface energy balance algorithms for land (SEBAL) model in Sanjiang Plain, Northeast China. Chinese Geographical Science 23:73–91 Esmi Jahromi M, Afzali SH (2014) Application of the HBMO approach to predict the total sediment discharge. Iranian J Sci Technol 38:123– 135 Fooladmand HR, Haghighat M (2007) Spatial and temporal calibration of Hargreaves equation for calculating monthly ETo based on PenmanMonteith method. Irrigation and Drainage 56:439–449 Ford CR, Hubbard RM, Kloeppel BD, Vose JM (2007) A comparison of sap flux-based evapotranspiration estimates with catchment-scale water balance. Agricultural and Forest Meteorology 145:176–185 George BA, Reddy B, Raghuwanshi N, Wallender W (2002) Decision support system for estimating reference evapotranspiration. J Irrig Drain Eng 128:1–10 Haddad OB, Afshar A, Mariño MA (2006) Honey-bees mating optimization (HBMO) algorithm: a new heuristic approach for water resources optimization. Water Resour Manag 20:661–680 Hargreaves GH (1975) Moisture availability and crop production. Trans. ASAE 18:980–984 Hargreaves GH, Samani ZA (1985) Reference crop evapotranspiration from ambient air temperature. American Society of Agricultural Engineers 1:96–99 Hargreaves GL, Hargreaves GH, Riley JP (1985) Irrigation water requirements for Senegal River basin. J Irrig Drain Eng 111:265–275 Hussein, AS (1999) Grass ET estimates using Penman-type equations in Central Sudan. J Irrig Drain Eng 125:324–329 Jabloun Md, Sahli A (2008) Evaluation of FAO-56 methodology for estimating reference evapotranspiration using limited climatic data: Application to Tunisia. Agric Water Manag 95:707–715 Jacovides C, Kontoyiannis H (1995) Statistical procedures for the evaluation of evapotranspiration computing models. Agric Water Manag 27:365–371 Jahromi ME, Ehsan M, Meyabadi AF (2012) A dynamic fuzzy interactive approach for DG expansion planning. Int J Electr Power Energy Syst 43:1094–1105 Jensen ME, Haise HR (1963) Estimating evapotranspiration from solar radiation. Proceedings of the American Society of Civil Engineers, Journal of the Irrigation and Drainage Division 89:15–41 Jensen D, Hargreaves G, Temesgen B, Allen R (1997) Computation of ETo under nonideal conditions. J Irrig Drain Eng 123:394–400 Jun F, Mingan S, Quanjiu W (2008) Comparison of many equations for calculating reference evapotranspiration in the Loess Plateau of China. Transactions of the Chinese Society of Agricultural Engineering 2008. doi:10.3969/j.issn.1002-6819.2008.3.019 Kisi O (2013) Comparison of different empirical methods for estimating daily reference evapotranspiration in Mediterranean climate. J Irrig Drain Eng 140:04013002 Landeras G, Ortiz-Barredo A, López JJ (2008) Comparison of artificial neural network models and empirical and semi-empirical equations
for daily reference evapotranspiration estimation in the Basque Country (Northern Spain). Agric Water Manag 95:553–565 Liu Y, de Jeu RA, van Dijk AI, Owe M (2007) TRMM-TMI satellite observed soil moisture and vegetation density (1998–2005) show strong connection with El Niño in eastern Australia. Geophys Res Lett 34:L15401 López-Urrea R, de Santa Olalla FM, Fabeiro C, Moratalla A (2006) Testing evapotranspiration equations using lysimeter observations in a semiarid climate. Agric Water Manag 85:15–26 Makkink G (1957) Testing the Penman formula by means of lysimeters. J Inst Water Eng 11:277–288 Marinakis Y, Marinaki M, Dounias G (2011) Honey bees mating optimization algorithm for the Euclidean traveling salesman problem. Inf Sci 181:4684–4698 Meyer, WS (1999) Standard reference evaporation calculation for inland, South Eastern Australia. CSIRO Land and Water Moritz RF, Southwick EE (1992) Bees as superorganisms: an evolutionary reality. Springer Verlag Oudin L, Hervieu F, Michel C, Perrin C, Andréassian V, Anctil F, & Loumagne C (2005) Which potential evapotranspiration input for a lumped rainfall–runoff model?: Part 2—Towards a simple and efficient potential evapotranspiration model for rainfall–runoff modelling. J Hydrol 303:290–306 Pereira AR, Pruitt WO (2004) Adaptation of the Thornthwaite scheme for estimating daily reference evapotranspiration. Agric Water Manag 66:251–257 Priestley C, Taylor R (1972) On the assessment of surface heat flux and evaporation using large-scale parameters. Mon Weather Rev 100: 81–92 Sabar NR, Ayob M, Kendall G, Qu R (2012) A honey-bee mating optimization algorithm for educational timetabling problems. Eur J Oper Res 216:533–543 Sabziparvar A-A, Tabari H, Aeini A, Ghafouri M (2010) Evaluation of class A pan coefficient models for estimation of reference crop evapotranspiration in cold semi-arid and warm arid climates. Water Resour Manag 24:909–920 Senay, G, Leake, S, Nagler, P, Artan, G, Dickinson, J, Cordova, J, Glenn, E (2011) Estimating basin scale evapotranspiration (ET) by water balance and remote sensing methods. Hydrol Process 25:4037–4049 Tabari H, Grismer ME, Trajkovic S (2013) Comparative analysis of 31 reference evapotranspiration methods under humid conditions. Irrig Sci 31:107–117 Thornthwaite CW (1948) An approach toward a rational classification of climate. Geogr Rev 38:55–94 Trajkovic S (2005) Temperature-based approaches for estimating reference evapotranspiration. J Irrig Drain Eng 131:316–323 Trajkovic S (2007) Hargreaves versus Penman-Monteith under humid conditions. J Irrig Drain Eng 133:38–42 Trajkovic S, Kolakovic S (2009a) Evaluation of reference evapotranspiration equations under humid conditions. Water Resour Manag 23: 3057–3067 Trajkovic S, Kolakovic S (2009b) Comparison of simplified pan-based equations for estimating reference evapotranspiration. J Irrig Drain Eng 136:137–140 Turc L (1961) Evaluation des besoins en eau d’irrigation, évapotranspiration potentielle. Ann Agron 12:13–49 Tyagi, NK, Sharma, DK, Luthra, SK (2003) Determination of evapotranspiration for maize and berseem clover. Irrig Sci 21:173–181 Valiantzas JD (2006) Simplified versions for the Penman evaporation equation using routine weather data. J Hydrol 331:690–702 Valiantzas JD (2012a) Simple ET forms of Penman’s equation without wind and/or humidity data. I: Theoretical Development Journal of Irrigation and Drainage Engineering 139:1–8 Valiantzas JD (2012b) Simple ET forms of Penman’s equation without wind and/or humidity data. II: comparisons with reduced set-FAO and other methodologies. J Irrig Drain Eng 139:9–19
Ahooghalandari M. et al. Valiantzas JD (2012c) Simplified reference evapotranspiration formula using an empirical impact factor for Penman’s aerodynamic term. J Hydrol Eng 18:108–114 Van Vreeswyk AME (2004) An inventory and condition survey of the Pilbara region, Western Australia. Department of Agriculture, Australia Vaughan PJ, Ayars JE (2009) Noise reduction methods for weighing lysimeters. J Irrig Drain Eng 135:235–240 Wilson KB, Hanson PJ, Mulholland PJ, Baldocchi DD, Wullschleger SD (2001) A comparison of methods for determining forest
evapotranspiration and its components: sap-flow, soil water budget, eddy covariance and catchment water balance. Agric For Meteorol 106:153–168 Yuce B, Packianather MS, Mastrocinque E, Pham DT, Lambiase A (2013) Honey bees inspired optimization method: the Bees Algorithm. Insects 4:646–662 Zhai L, Feng Q, Li Q, Xu C (2010) Comparison and modification of equations for calculating evapotranspiration (ET) with data from Gansu Province, Northwest China Irrig Drain 59:477–490
