Reference evapotranspiration (ETo) is widely used for agricultural, hydrological, and environmental studies for sustainable water use. It is estimated by different methods from measurement though lysimeters to mathematical modeling using climatic and geographical variables [1]. Numerous equations have been developed to estimate ETo at different time scales across the globe. Hargreaves and Samani [2] proposed temperature based ETo equation which was world wise used with variable accuracy [3-5]. Other ETo equations [6-19], the authors provided variable performance under different climatic conditions and locations. The Penman-Monteith equation [16] in its standardized form was revealed the most accurate ETo method under all climatic conditions across the globe [16,20-30]. While different ETo methods are providing ETo estimated with relatively good accuracy but nonadapted to all the climatic conditions, their calibration to local climatic conditions offers the opportunity to improve the performance of these ETo equations under weather conditions different from the conditions under which they were developed. Droogers and Allen [31] reported suitability of very few ETo equation to the Iranian environment. Djaman et al. [32] calibrated the Trabert, Mahringer, Albrecht, and two of the Valiantzas equations to the Sahelian conditions in the Senegal River Valley. The Hargreaves ETo equation was calibrated under different climatic conditions [33]. Valiantzas [34] had lately proposed some new simplified forms of the Penman-Monteith ETo equation and which have been used for daily ETo estimation with variable performance related to the local climatic conditions. While some of these equations showed good accuracy in ETo estimations under some climatic conditions, others need calibration to improve their performance [32-35]. One of the Valiantzas ETo equations was shown suitable for ETo estimation across Burkina Faso [36,37], Tanzania, Kenya and Uganda [38,39]. Ahooghalandari et al. [35] indicated good performance of the calibrated forms of two of the Valiantzas equations to the Pilbara region of Western Australia. The Valiantzas ETo equations have also been tested by Valipour [40,41] and Kisi [42] with variable performance. Different forms for the Valiantzas ETo equation with full dataset gave different performance in Iran related to the geographical locations, wind speed range and the magnitude of the shortwave incoming solar radiation [40]. While, Valiantzas [34] has suggested different equations as a function of the availability of climatic data, one with full climatic variables showed its accuracy in Uganda for the 1992-2012 period at 16 weather stations with R2 of 0.9975 and simple linear regression slope of 1.0055 for the pooled data set and RMSE varying from 0.06 to 0.09 mm/day and MBE from 0.008 to 0.064 mm/day across the locations [39]. This equation was revealed adapted to the studied region and could be used without any calibration. The objective of this study was to evaluate the most performing Valiantzas equation in comparison with the Penman-Monteith equation under humid, sub-humid and semiarid conditions and across Africa for its regional adaptability.

Study area and meteorological data used

This study covers different climatic zones from the humid to semiarid climatic conditions in Africa. Meteorological data were collected from ten continental African countries and Madagascar (Grand African Island) covering different climatic conditions and different periods: Benin (1980-2010), Burkina Faso (1998-2012), Cameroun (1980- 2010), Ghana (1980-2010), Kenya (1998-2012), Madagascar (1980- 2010), Niger (1998-2012), Nigeria (1998-2012), Senegal (1950-2012), Tanzania (1998-2012). Daily average solar radiation (Rs), minimum temperature (Tmin), maximum temperature (Tmax), minimum relative humidity (RHmin), maximum relative humidity (RHmax) and wind speed (u) were monitored at different weather stations in these countries with long-term average climatic conditions summarized in Tables 1 and 2.

Table 1: Geographic coordinates and annual average climatic variables of the weather observatories in the continental Africa.

Table 2: Geographic coordinates and annual average climatic variables of the weather observatories in Madagascar (Grand African Island).

Reference evapotranspiration equations

Penman-Monteith model (PM-ETo): Daily grass-reference ET was computed using the standardized ASCE form of the Penman- Monteith (PM-ETo) equation [20]:

(1)

whee: ETo is the reference evapotranspiration (mm day^-1), Δ is the slope of saturation vapor pressure versus air temperature curve (kPa °C^-1), Rn is the net radiation at the crop surface (MJ m^-2 d^-1), G is the soil heat flux density at the soil surface (MJ m^-2 d^-1), T is the mean daily air temperature at 1.5-2.5 m height (C), u₂ is the mean daily wind speed at 2 m height (m s^-1), es is the saturation vapor pressure at 1.5-2.5 m height (kPa), ea is the actual vapor pressure at 1.5-2.5 m height (kPa), es-ea is the saturation vapor pressure deficit (kPa), γ is the psychrometric constant (kPa °C^-1), Cn and Cd are constants with values of 900°C mm s³ Mg^-1 d^-1 and 0.34 s m^-1, respectively. The procedure developed by Turc [18] was used to compute the needed parameters.

Valiantzas reference evapotranspiration equation Valiantzas

(2)

Where, Tmax, Tmin, and Tmean are daily maximum, minimum, and mean air temperature ( °C), respectively; RH is daily relative humidity (%); Rs is solar radiation (MJ m^-2 day^-1), Ra is extraterrestrial radiation (MJ m^-2 day^-1); and α=0.25, φ is the latitude of the weather station in radians, z is the elevation (m) of the weather station [34].

Evaluation criteria

Daily ETo estimates using the Vlaiantzas equation were compared with the daily ETo estimates by the Penman-Monteith equation using graphics and simple linear regression, root mean squared error (RMSE), relative error (RE), mean bias error (MBE), and mean absolute error (MAE).

(3)

(4)

(5)

(6)

Where, ETo Vali is the estimated ETo with the Valiantzas’ ETo equation; and EToPMi is ETo estimated with PM-ETo model, at the ith data point and n is the total number of data points.

The evaluation of the Valiantzas’ ETo equation versus Penman- Monteith equation at country level is presented in Figure 1 and the statistics are summarized in Table 3. Perfect agreement between the two equations was shown in the Sahelian environment in Niger and Nigeria, and in humid climatic conditions in Tanzania, and Kenya. Valinantzas’ ETo equation underestimated daily ETo at high daily ETo values above 10 mm/day in the sub-humid climatic conditions in Benin, Cameroun, Ghana, and in the semiarid climate in Senegal. The Valiantzas’ equation showed good agreement with Penman-Monteith equation with regression slopes that varied from 0.99 to 1.02 and the coefficient of determination varying from 0.98 to 0.99. The RMSE was low and ranged from 0.06 to 0.22 mm/day with the highest RMSE value obtained in Benin and the lowest value in Kenya, and Tanzania. At country level, the percent error values were acceptable and lower than 4.5%, Table 2 showing the very low error in the estimates of daily ETo by Valiantzas’ equation. Moreover, very low MBE and MAE values were obtained and varied from -0.09 to 0.13 mm/day and from 0.04 to 0.15 mm/day, respectively (Table 3). The RMSE, PE, MBE and MAE averaged 0.10 mm/day, 1.95%, 0.02 mm/day and 0.08 mm/day, respectively.

Table 3: Statistical Indexes for the evaluation of Valiantzas’ reference evapotranspiration equation in 10 countries covering humid and semiarid climatic conditions.

Figure 1: Valiantzas vs Penman-Monteith equations for the pooled data at country level.

At station level, the results of the performance evaluation of Valiantzas’ ETo equation are summarized in Table 3. Very good agreement was found between Valiantzas and Penman-Monteith equations. Very low RMSE was obtained and varied from 0.03 to 0.27 mm/day. Moreover, the percent error PE was also very low and varied from 0.87 to 5.46% only. MBE varied from -0.09 to 0.23 mm/ day and MAE ranged from 0.03 to 0.23 mm/day (Tables 4 and 5). PE values lower than 2% were obtained at 75% of the weather stations. The poorest performance of Valiantzas’ equation was revealed across Burkina Faso with the PE values varying from 1.94 to 5.46%. However, the highest value de RMSE of 0.27 mm/day was registered at Lagdo in Cameroun under humid climate and the lowest value of 0.05 mm/day were registered at Songea in Tanzania. Average RMSE of all 61 weather stations was as low as 0.10 mm/day and the PE averaged only 1.95%. Average MBE and MAE values were 0.02 and 0.008 mm/day, respectively. The results of this study are in agreement with Gelcer et al. [43] who indicated that the Valiantzas’ method performed the best among six ETo equations evaluated at 92 automated weather stations in Florida, Georgia, and North Carolina (Southeast USA). The authors reported very low percent error at 92 weather stations under humid climate in Florida, Georgia, and North Carolina. Similar results were reported by Kisi [42] in Turkey, Valipour [41] in Iran, Ahooghalandari et al. [35] in Australia. The Valiantzas’ ETo equation performed well at 16 weather stations in Uganda for the period of 1992-2012 with a regression slope and R2 of the pooled data of 1.0055 and of 0.9975, respectively, relative to the Penman-Monteith ETo equation. Moreover, RMSE varied from 0.05 to 0.09 mm/day; MBE varied from 0.008 to 0.05 mm/day and the regression slope varied from 1.00 to 1.02 indicating the applicability of the Valiantzas ETo equation across Uganda [39]. McShea [44] reported good agreement between the Valiantzas ETo estimated and Penman-Monteith ETo estimates in the Mountainous cold regions in Colorado (USA). Shalamzari and Mohammadi [45] reported that the Valiantzas’ ETo equation showed the best performance under humid cold climate region of Chaharmahal and Bakhtiary province, Borujen, Shahrekord, Koohrang and Lordegan in Iran, for the period of 1994- 2015. VAlipour [41] reported the Valiantzas ETo equation under this study to be the most precise method for the East and South Iran. Adversely, Peng et al. [46] reported the valiantzas ETo equation to have the worst performance than the other ETo equations under their study in different sub-regions of the mainland China.

Table 4: Statistical Indexes for the evaluation of Valiantzas reference evapotranspiration equation at the individual weather stations in 10 countries covering humid and semiarid climatic conditions in the continental Africa.

Table 5: Statistical Indexes for the evaluation of Valiantzas reference evapotranspiration equation at the individual weather stations in Madagacar (Grand Africa Island).

Similar to the adaptability of Valiantzas equation, Hargreaves equation is widely used for daily or monthly ETo estimation with variable degree of performance in Spain [47], in Italy [4,48], in South Korea [49], in Iran [50-52], in Canada [53], in Senegal, Kenya, and Tanzania [5,38], in Burkina Faso [37]; in China [46,54]. However, climatic variables required by the Hargreaves equation are very limited compared to the Valiantzas equation which required similar parameters as the Penman-Monteith equation. This increases the usefulness and the applicability of the Hargreaves equation across the globe, mostly under the environments where only very few climatic parameters are recorded like in the developing countries similar to the African countries.

The MBE value ranged between -0.09 and 0.13 mm/day with the highest value in Burkina Faso and the lowest value in Senegal both under the semiarid climate while the overall average MBE was -0.026 mm/day with slight underestimation of the daily ETo by the Valianstzas equation equivalent to a deficit of 0.26 m3/ha of irrigation and or rainfall water. With %PE range of 1.3-4.3 which is relatively low, the Valiantzas equation could be applied at regional and global levels without putting crop under water stress. Thus, crops support up to 25% of deficit irrigation application without any yield reduction in maize [55-58], in soybean [59,60], and in sweet potato [61]. Even if some online resources exist to help estimating the Penman-Monteith ETo values, the accessibility and internet network coverage and intensity limit the use of such resources. Therefore, the use of the Valiantzas’ equation might be much easier in most developing countries for aforementioned reasons in addition to numerous intermediate calculations when using the Penman-Monteith equation and which are subjected to some errors that can be detrimental to irrigation scheduling and water management under agricultural, hydrological and environmental studies and projects. The Valiantzas’ ETo equation could therefore be an alternative to the Penman-Monteith ETo equation when full climatic dataset is available. However, Penman-Monteith ETo equation should overall still is adopted and applied with caution when all the required parameters by Valiantzas equation are available.

The performance of Valiantzas’ reference evapotranspiration equation was evaluated under this study using 61 weather stations across 10 African countries covering humid, sub-humid and semiarid climates for the period of 1980-2012. The results showed good agreement between the Penman-Monteith and the Valiantzas’ equation at all weather station with RMSE values that varied from 0.03 to 0.27 mm/day, percent error PE from 0.87 to 5.46%, MBE from -0.09 to 0.23 mm/day and MAE from 0.03 to 0.23 mm/day. The Valiantzas’ ETo equation could be an alternative to the Penman-Monteith equation without calibration to local humid, sub-humid and semiarid climatic conditions.

1. Mendonça JC, Sousa EF, de Bernardo S, Dias GP, Grippa S (2003) Comparison of estimation methods of reference crop evapotranspiration (ETo) for Northeren Region of Rio de Janeiro State, Brazil. Revista Brasileira de Engenharia Agrícolae Ambiental 7: 275-279.
2. Hargreaves GH, Samani ZA (1985) Reference crop evapotranspiration from temperature. Applied Engineering in Agriculture 1: 96-99.
3. Bautista F, Bautista D, Delgado-Carranza C (2009) Calibration of the equations of Hargreaves and Thornthwaite to estimate the potential evapotranspiration in semi-arid and sub-humid tropical climates for regional applications. Atmósfera 22: 331-348.
4. Berti A, Tardivo G, Chiaudani A, Rech F, Borin M (2014) Assessing reference evapotranspiration by the Hargreaves methodin north-eastern Italy. Agricultural Water Management 140: 20-25.
5. Djaman K, Balde AB, Sow A, Muller B, Irmak S, et al. (2015) Evaluation of sixteen reference evapotranspiration methods under sahelian conditions in the Senegal River Valley. Journal of Hydrology: Regional Studies 3: 139-159
6. Irmak S, Irmak A, Jones JW (2003) Solar and net radiation-based equations to estimate reference evapotranspiration in humid climates. Journal of Irrigation and Drainage Engineering 129: 336-347.
7. Blaney HF, Criddle WD (1962) Determining consumptive use and irrigation water requirements. USDA Technical Bulletin 1275, US Department of Agriculture, Beltsville.
8. Trabert W (1896) Recent observations on evaporation rate. Meteorologist Z  13: 261-263.
9. Penman HL (1963) Vegetation and hydrology. Soil Science 96: 357.
10. Dalton J (1802) Experimental essays on the constitution of mixed gases: on the force of steam or vapor or other liquids in different temperatures, both in a Torricelli vacuum and in air, on evaporation, and on expansion of gases by heat. Memoirs of the Literary and Philosophical Society of Manchester 5: 536-602.
11. Makkink GF (1957) Testing the Penman formula by means of lysimeters. Journal of the Institution of Water Engineers 11: 277-288.
12. Abtew W (1996) Evapotranspiration measurements and modeling for three wetland systems in south Florida. Journal of American Water Resources Association 32: 465-473.
13. Jensen ME, Haise HR (1963) Estimating evapotranspiration from solar radiation. Proceedings of the American Society of Civil Engineers, J Irrig Drain Div.89: 15-41.
14. Schendel U (1967) Vegetations wasserverbrauch und wasserbedarf Habilitation. Kiel 137
15. Thornthwaite CW (1948) An approach towards a rational classification of climate. Geogr Revue 38: 55-94.
16. Allen RG, Pereira LS, Raes D, Smith M (1998) Crop evapotranspiration: Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper No. 56 300.
17. Hansen S (1984) Estimation of potential and actual evapotranspiration. Nordic Hydrology 15: 205-212.
18. Turc L (1961) Water requirements assessment of irrigation, potential evapotranspiration: simplified and updated climatic formula. Annales Agronomiques 12: 13-49.
19. Priestley CHB, Taylor RJ (1972) On the assessment of surface heat flux and evaporation using large-scale parameters. Monthly Weather Review 100: 81-92.
20. Irmak S, Howell TA, Allen RG, Payero JO, Martin DL (2005) Standardized ASCE Penman-Monteith: Impact of sum-of-hourly vs. 24-hour timestep computations at reference weather station sites. Transactions of the ASAE 48: 1063-1077.
21. Utset A, Farre I, Martinez-Cob A, Cavero J (2004) Comparing Penman-Monteith and Priestley-Taylor approaches as reference evapotranspiration inputs for modeling maize water use under Mediterranean conditions. Agricultural Water Management 66: 205-219.
22. López-Urrea R, Martín de Santa Olalla F, Fabeiro C, Moratalla A (2006) Testing evapotranspiration equations using lysimeter observations in a semiarid climate. Agricultural Water Management 85: 15-26.
23. Bodner G, Loiskand W, Kaulm H (2007) Cover crop evapotranspiration under semi-arid conditions using FAO dual crop coefficient method with water stress compensation. Agricultural Water Management 93: 85-98.
24. Irmak S, Irmak A, Howell TA, Martin DL, Payero JO, et al. (2008) Variability analyses of alfalfa-reference to grass-reference evapotranspiration ratios in growing and dormant seasons. Journal of Irrigation and Drainage Engineering 134: 147-159.
25. Jabloun M, Sahli A (2008) Evaluation of FAO-56 methodology for estimating reference evapotranspiration using limited climatic data application to Tunisia. Agricultural Water Management 95: 707-715.
26. Xing Z, Chow L, Meng F, Rees HW, Monteith J, et al. (2008) Testing reference evapotranspiration estimation methods using evaporation pan and modeling in Maritime Region of Canada. Journal of Irrigation and Drainage Engineering 134: 417-424.
27. Trajkovic S, Kolakovic S (2009) Evaluation of reference evapotranspiration equations under humid conditions. Water Resource Management 23: 3057-3067.
28. Martinez CJ, Thepadia M (2010) Estimating reference evapotranspiration with minimum data in Florida, USA. Journal of Irrigation and Drainage Engineering 136: 494-501.
29. Tabari H, Grismer M, Trajkovic S (2011) Comparative analysis of 31 reference evapotranspiration methods under humid conditions. Irrigation Science 31: 107-117.
30. Xystrakis F, Matzarakis A (2011) Evaluation of 13 empirical reference potential evapotranspiration equations on the island of Crete in southern Greece. Journal of Irrigation and Drainage Engineering 137: 211-222.
31. Droogers P, Allen RG (2002) Estimating reference evapotranspiration under inaccurate data conditions. Irrigation and drainage systems 16: 33-45.
32. Djaman K, Tabari H, Balde AB, Diop L, Futakuchi K, et al. (2016) Analyses, calibration and validation of evapotranspiration models to predict grass reference evapotranspiration in the Senegal River Delta. Journal of Hydrology: Regional Studies 8: 82-94.
33. Hargreaves GH, Allen RG (2003) History and evaluation of Hargreaves evapotranspiration equation. Journal of Irrigation and Drainage Engineering 129: 53-63.
34. Valiantzas JD (2013) Simplified reference evapotranspiration formula using an empirical impact factor for penman’s aerodynamic term. Journal of Irrigation and Drainage Engineering 18: 108-114.
35. Ahooghalandari M, Khiadani M, Jahromi ME (2016) Calibration of Valiantzas’ reference evapotranspiration equations for the Pilbara region, Western Australia. Theoretical and Applied Climatology 1-12.
36. Djaman K, Irmak S, Kabenge I, Futakuchi K (2016) Evaluation of the FAO-56 Penman-Monteith model with limited data and the Valiantzas’ models for estimating reference evapotranspiration in the Sahelian conditions. Journal of Irrigation and Drainage Engineering 142.
37. Ndiaye PM, Bodian A, Diop L, Djaman K (2017) Evaluation of twenty methods for estimating daily reference evapotranspiration in Burkina Faso. Physio-Geo 11: 129-146.
38. Djaman K, Irmak S, Futakuchi K (2016) Daily Reference Evapotranspiration estimation under Limited Data in Eastern Africa. J Irrig Drain Eng 143.
39. Djaman K, Rudnick D, Mel VC, Mutiibwa D, Diop L, et al. (2017) Evaluation of the Valiantzas’ simplified forms of the FAO-56 Penman-Monteith Reference Evapotranspiration model under humid climate. Journal of Irrigation and Drainage Engineering 148.
40. Valipour M (2014) Investigation of Valiantzas’ evapotranspiration equation in Iran. Theoretical and Applied Climatology 121: 1-2.
41. Valipour M (2015) Importance of solar radiation, temperature, relative humidity, and wind speed for calculation of reference. Archives of Agronomy and Soil Science 6: 239-255.
42. Kisi O (2014) Closure to Comparison of Different Empirical Methods for Estimating Daily Reference Evapotranspiration in Mediterranean Climate. Journal of Irrigation and Drainage Engineering 141.
43. Gelcer EM, Fraisse CW, Sentelhas PC (2010) Evaluation of Methodologies to Estimate Reference Evapotranspiration in Florida. Proceedings of the Florida State Horticultural Society 123: 189-195.
44. McShea VM (2017) Identifying climate related hydrologic regime change in Mountainous watersheds. (Doctoral dissertation, Colorado School of Mines. Arthur Lakes Library).
45. Kobra SK, Mohammadi AS (2016) Comparison of different empirical methods for estimating daily reference evapotranspiration in the humid cold climate. Journal of water science engineering 6: 21-38.
46. Peng L, Li Y, Feng H (2017) The best alternative for estimating reference crop evapotranspiration in different sub-regions of mainland China. Sci Rep 7: 5458.
47. Maestre-Valero JF, Martínez-Álvarez V, González-Real MM (2013) Regionalization of the Hargreaves coefficient to estimate long-term reference evapotranspiration series in SE Spain. Spanish Journal of Agricultural Research 11: 1137-1152.
48. Mendicino G, Senatore A (2013) Regionalization of the Hargreaves coefficient for the assessment of distributed reference evapotranspiration in southern Italy. Journal of Irrigation and Drainage Engineering 139: 349-362.
49. Jung CG, Lee DR, Moon JW (2016) Comparison of the Penman-Monteith method and regional calibration of the Hargreaves equation for actual evapotranspiration using SWAT-simulated results in the Seolma-cheon basin, South Korea. Hydrological Sciences Journal 61: 793-800.
50. Fooladmand HR, Haghighat M (2007) Spatial and temporal calibration of Hargreaves equation for calculating monthly reference evapotranspiration based on Penman–Monteith method. Irrigation and Drainage 56: 439-449.
51. Tabari H, Talaee PH (2011) Local calibration of the Hargreaves and Priestley-Taylor equations for estimating reference evapotranspiration in arid and cold climates of Iran based on the Penman-Monteith model. Journal of Hydrologic Engineering 16: 837-845.
52. Heydari MM, Heydari M (2014) Calibration of Hargreaves-Samani equation for estimating reference evapotranspiration in semiarid and arid regions. Archives of Agronomy and Soil Science 60: 695-713.
53. Sentelhas PC, Gillespie TJ, Santos EA (2010) Evaluation of FAO Penman–Monteith and alternative methods for estimating reference evapotranspiration with missing data in Southern Ontario, Canada. Agricultural Water Management 97: 635-644.
54. Gao X, Peng S, Xu J, Yang S, Wang W (2015) Proper methods and its calibration for estimating reference evapotranspiration using limited climatic data in Southwestern China. Archives of Agronomy and Soil Science 61: 415-426.
55. Djaman K, Irmak S, Rathje WR, Martin D, Eisenhauer D (2013) Maize evapotranspiration, yield production functions, biomass, grain yield, harvest index, and yield response factors under full and limited irrigation. Transactions of the ASABE 56: 273-293.
56. Domínguez A, de Juan JA, Tarjuelo JM, Martínez RS, Martínez-Romero A (2012) Determination of optimal regulated deficit irrigation strategies for maize in a semi-arid environment. Agricultural Water Management 110: 67-77.
57. Irmak S, Specht JE, Odhiambo LO, Rees JM, Cassman KG (2015) Soybean yield, water productivity, evapotranspiration and soil-water extraction response to subsurface drip irrigation. Transactions of the ASABE 57: 729-748.
58. Irmak S, Djaman K, Rudnick DR (2016) Effect of full and limited irrigation rate and frequency on subsurface drip-irrigated maize evapotranspiration, yield, water use efficiency and yield response factors. Irrigation Science 34 : 271-286.
59. Rosadi RAB, Afandi MS, Oti K, Adomako JT (2007) The Effect of Water Deficit in Typical Soil Types on the Yield and Water Requirement of Soybean (Glycine max [L.] Merr.) in Indonesia. Japan Agricultural Research Quaterly. 41: 47-52.
60. Sandhu RK (2016) Effect of Soybean Seeding Rates on Yield, Evapotranspiration, Crop Coefficients and Crop Water Productivity under Rainfed, Deficit and Full Irrigation Regimes Using Subsurface Drip Irrigation System. Master’s Thesis, University of Nebraska-Lincoln, USA.
61. Martin M, Miller D (1983) Variations in responses of potato germplasm to deficit irrigation as affected by soil texture. American Potato Journal 60: 671-683.