FAO-56 methodology for determining water ...

Theor Appl Climatol DOI 10.1007/s00704-013-0972-3

ORIGINAL PAPER

FAO-56 methodology for determining water requirement of irrigated crops: critical examination of the concepts, alternative proposals and validation in Mediterranean region Nader Katerji & Gianfranco Rana

Received: 8 January 2013 / Accepted: 8 July 2013 # Springer-Verlag Wien 2013

Abstract The present study evaluates firstly the ability of the FAO-56 methodology, based on the two-step approach “reference evapotranspiration (ET0)—crop coefficient (Kc),” to accurately determine the actual evapotranspiration (ET) of irrigated crops and proposes, secondly, the alternative approaches for improving this determination. The FAO-56 methodology is supported by two hypotheses: (1) ET0 represents all effects of weather and (2) Kc varies predominately with specific crop characteristics and only marginally with climate, which enables the transfer of Kc standard values among locations and climates. On the base of the theoretical analysis and experimental observations, a critical examination of the previous hypotheses demonstrates that they are not verified by reality. The first hypothesis is not verified for two reasons: (a) The formulation adapted by the Penman– Monteith equation and proposed in FAO-56 methodology for calculating ET0 uses climatic variables determined at a 24-h average scale. However, in principle it is only valid in permanent regime, in other words at least on an hourly scale. (b) The FAO-56-proposed formulation attributes a constant value to the canopy resistance of the reference surface; but in reality, this resistance is variable in relation to the climatic variables. The second hypothesis, concerning the two-step approach, is also not verified because the values of Kc largely vary in relation to climatic variables (radiation, air vapour pressure deficit and wind speed). This fact does not support the possibility of the transferability of Kc values into locations

N. Katerji INRA—Unité Mixte de Recherche Environnement et Grandes Cultures, 78850 Thiverval-Grignon, France e-mail: [email protected] G. Rana (*) Consiglio per la Ricerca e la Sperimentazione in Agricoltura (CRA)—Research Unit on Agriculture in Dry Environments, via C. Ulpiani, 5, 70125 Bari, Italy e-mail: [email protected]

where the local conditions deviate from the conditions where the adjusted values of Kc were determined. The weakness of the ET estimation, observed on several crops cultivated in the Mediterranean region, through the application of the FAO-56 methodology, is the result of errors accumulation, associated with that affects the determination of either ET0 or Kc. The present study underlines the advantage of using a one-step approach in the calculation of ET, since it is based on fewer computation steps and, consequently, on fewer error sources than the two-step model. Two models adopting this approach are proposed and validated, one of which can be considered as operational, i.e. it only needs standard meteorological data as input. The use of these models enables an improvement of the ET estimation. This objective is a key component of any strategy to improve agricultural water management in Mediterranean region.

1 Introduction In 1990, the Food and Agricultural Organisation (FAO) organised an expert consultation (Smith et al. 1992) concerning the revision of the FAO-24 publication on crop water requirements by Doorenbos and Pruitt (1977). The well-known FAO-56 publication, on crop evapotranspiration (i.e. the water requirement) by Allen et al. (1998), is the final product of this revision project. Following Jensen (2010), this last publication puts a century of progress of the estimation of crop evapotranspiration into concrete form. In this publication (Allen et al. 1998) retained the two-step approach “reference evapotranspiration (ET0)—crop coefficient (Kc)” previously adopted by Doorenbos and Pruitt in the FAO-24 publication. Following this approach, two steps (van Wijk and de Vries 1954; Jensen 1968; Allen et al. 1998) are necessary for determining the evapotranspiration (ET) of irrigated crops:

N. Katerji, G. Rana

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First, to calculate the reference evapotranspiration ET0. This term represents all effects of weather (Allen et al. 2006). This is the first hypothesis on which the two-step approach is based. Second, to correct the calculated value of ET0 with the values of the crop coefficient Kc (ET=ET0 ×Kc). This coefficient varies primarily with specific crop characteristics and only somewhat with climate, so this enables the transfer of standard Kc values among locations and climates (Allen et al. 2006). This is the second hypothesis required for the two-step approach.

However, in the FAO-56 publication, Allen et al. (1998) proposed new methods with respect to FAO-24 that primarily concerned the following: –

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The determination of ET0. The formulas previously used by Doorenbos and Pruitt (1977) (modified Penman, Blaney–Criddle, Radiation) were abandoned and substituted with a formula based on the Penman– Monteith (PM) equation, as elaborated by Monteith (1965). This last equation was adapted and validated by the authors, for the irrigated reference grass, in a previous study (Allen et al. 1989). The determination of the coefficient Kc. Two definitions of the Kc curve during the growth cycle were provided in Allen et al. (1998). The first, referred to as the “single Kc approach,” which was already adopted by Doorenbos and Pruitt (1977), integrates the relationship between ET and ET0 into a time average. The single Kc approach was recommended by Allen et al. (1998) for irrigation practice. The second, usually referred to as the “dual approach,” is the algebraic sum of a basal crop coefficient and a soil evaporation coefficient. This last approach is recommended for research work. The possibility of determining the crop evapotranspiration in non-standard conditions (soil drought, soil salinity and combined salinity–drought). For each of these three conditions, Allen et al. (1998) proposed calculating a particular water stress factor Ks, which is used to correct the ET values calculated for standard irrigated crops.

A large body of literature has been produced to validate the calculation of ET0, Kc and ET in different sites following the FAO-56 methodology. Considering only studies carried out in the Mediterranean region, the works by Rana et al. (1994), Steduto et al. (1996), Todorovic (1999), Ventura et al. (1999), Steduto et al. (2003), Lecina et al. (2003), Perez et al. (2006), Katerji and Rana (2006), Katerji and Rana (2011) and Espadafor et al. (2011) all finalised to calculate reference evapotranspiration ET0. The works performed by Ferreira et al. (1996), Beyazgül et al. (2000), Casa et al. (2000), Lascano (2000), Ferreira and Carr (2002),

Hamidat et al. (2002), Ayars et al. (2003), Williams et al. (2003), Johnson et al. (2004), Rinaldi and Rana (2004), Testi et al. (2004), Villalobos et al. (2004), Yunusa et al. (2004), Amayreh and Al-Abed (2005), Hanson and May (2005), Kar and Verma (2005), Karam et al. (2005), Lovelli et al. (2005), Orgaz et al. (2005), Parkes et al. (2005), Rana et al. (2005), Vu et al. (2005), Vazquez et al. (2005), Williams and Ayars (2005), De Medeiros et al. (2006), Ortega-Farias et al. (2006), Paço et al. (2006), Martinez-Cob (2007), Er-Raki et al. (2008), Barbagallo et al. (2009), De Tar (2009), Er-Raki et al. (2009), Favati et al. (2009), Lopez-Urrea et al. (2009), Manuel Casanova et al. (2009), Ko et al. (2009), Piccinni et al. (2009), Sahin et al. (2009), Villalobos et al. (2009), Čereković et al. (2010) and Yacoubi et al. (2010) were primarily devoted to the determination of Kc and ET. Furthermore, the works concerning the evaluation of the stress coefficient Ks are still rare and do not allow us to determine the quality of this estimations. The few studies regarding the coefficient Ks in drought conditions (Howell et al. 2004) or salinity (Katerji et al. 2011a) suggest that more research is needed for assessing this water stress factor under non-standard conditions. The aim of the present study is, firstly, to evaluate the ability of the FAO-56 methodology to accurately determine the ET of irrigated crops and, secondly, to propose alternative approaches for improving this determination. To achieve these objectives, we adopted a procedure including three steps: (1) to realise a critical examination of the robustness of the two hypotheses on which the two-step approach is based, starting from a theoretical analysis associated with experimental observations; (2) to use this analysis for interpreting the observed differences between the calculated values of ET0, Kc and ET and the measured values, for several crops in Mediterranean region during multi-local and multi-annual experiments; (3) to propose and validate alternative approaches for determining the ET of irrigated crops, in order to overcome the faults of the two-step approach. We restricted the experimental study to the Mediterranean region because its particular water resources conditions. In fact, the sectorial analysis of water use in this region shows that 72 % of the available water is used for agricultural purposes (Hamdy and Katerji 2006). Water resources are becoming rare, mainly in southern countries of this region (Margat 2008; Gao and Giorgi 2008; Vitale et al. 2010). Moreover, water is wasted in this countries and caused salinity pollution (Wild 2003; Katerji et al. 2003; Ceccarelli et al. 2004) because the farmers allocated a large amount of water, exceeding crop requirements, for all winter and summer crops (Shideed et al. 2005). Their crops are overirrigated by 30 to 50 % (Hamdy and Katerji 2006; Perego et al. 2012). Agriculture, the main consumer of freshwater in the Mediterranean region, is currently faced with the challenge

FAO-56 methodology for determining water requirement

of new approach to water resource management that insures the protection of water resources and their integrity. In this region, saving water by controlling water supply, through better determination of crop water requirements, is a key component of any strategy to improve water management in agriculture (Katerji et al. 2008; Torres et al. 2011).

crop, rc is the bulk canopy resistance in second per metre and ra is the aerodynamic resistance in second per metre between crop and the reference point z. For practical purposes, the determination of ET by the PM equation for an irrigated grass surface cultivated near a standard weather station requires the following: –

2 The determination of ET0 following the methodology proposed by FAO-56 After Allen et al. (1998), the reference surface should closely resemble an extensive surface of green grass, with an assumed uniform height of 0.12 m, an albedo of 0.23 and a fixed surface resistance. This surface is actively growing, completely shading the ground and has adequate water. The requirement that the crop surface should be extensive and uniform results from the assumption that all fluxes are one-dimensional and upwards in direction, without advection. These authors recommend the use of the PM equation for calculating the values of ET0. In general, the determination of ET0 has two aims: – –

In agronomy, the computation of ET0 is the first step for calculating the crop ET by the two-step approach, as previously stated. In hydrology, ET0 is one of the most important hydrological components for improving the water delivery strategy and assessing the hydrological impacts of climate change (Torres et al. 2011; Zhang et al. 2011).

2.1 The Penman–Monteith (PM) equation The PM equation was originally formulated for cultivated crops (Monteith 1965). It is only valid in a permanent regime, i.e. on a time scale from few minutes to 1 h, and for large enough surfaces, for which advection effects can be counted (Perrier 1975a, b, c; Brutsaert 1982; Stull 1988). The actual crop evapotranspiration λE (in watts per square metre) expressed in energy terms as latent heat flux is written as: . ΔA þ ρcp D ra  .  λE ¼ ð1Þ Δ þ γ 1 þ rc ra where A=Rn −G is the available energy in watts per square metre, λ is the latent heat of evaporation in Joule per kilogramme, ρ is the air density in kilogramme per cubic metre, Δ is the slope of the saturation pressure deficit versus temperature function in kilopascal per degree Celsius, γ is the psychrometric constant in kilopascal per degree Celsius, cp is the specific heat of moist air in Joule per kilogramme per degree Celsius, D is the vapour pressure deficit of the air in kilopascal measured at the reference point z above the

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The determination of weather variables A, D and the air temperature Ta (degree Celsius). These variables are usually measured directly by the meteorological stations, using a standard setup 2 m above the soil surface (the variable Ta) or calculated starting from other standard climate variables. In particular, the variable A is calculated either with global radiation or sunshine duration, and the variable D is calculated with the relative humidity RH and Ta. The determination of the aerodynamic resistance ra, as a function of the wind speed u (in metre per second) measured 2 m above the surface by standard meteorological stations. The determination of the canopy resistance rc through a process as detailed in the following.

2.1.1 The computation of the aerodynamic resistance ra For determining the resistance ra of the reference surface, Allen et al. (1998) proposed the following formulation:     zm −d zh −d ln ln z0m z0h ð2Þ ra ¼ 2 k uz where zm and zh are the measurement of the heights of wind speed and humidity respectively, z0m and z0h are the roughness lengths for the momentum and heat and water vapour transfer respectively, uz is the wind speed measured at height z and k is the von Kármán constant. However, because the height hc for the reference surface is considered to be constant (0.12 m), all the parameters in Eq. (2) are constant, i.e.:  .  d ¼ 2 3 ⋅hc ð3Þ

z0m ¼ 0:123⋅hc

ð4Þ

z0h ¼ 0:1⋅z0 m 0m

ð5Þ

By introducing these constant values into Eq. (2), the resistance ra for the reference surface can be calculated by the following relationship (Allen et al. 1998): ra ¼

208 uz

ð6Þ

N. Katerji, G. Rana

Typically, uz is the wind speed measured 2 m above the surface by standard meteorological stations.

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The first equation allows the calculation on a daily scale (in millimetre per day):

2.1.2 The determination of the resistance rc For determining the resistance rc, Allen et al. (1989, 1998) adopted the formulation proposed by Monteith et al. (1965), which was first validated using oat crops in England. This formulation can be written as follows: rc;daily ¼

rs LAI

ð7Þ

with rs being the stomatal resistance, averaged on an hourly scale per leaf surface unit, measured between 9 and 15 h; LAI is the Leaf Area Index. For adapting Eq. (7) to irrigated grass, Allen et al. (1989) introduced the following three simplifying hypotheses: –

– –

ET0 ¼

900 u2 D T a þ 273 Δ þ γ ð1 þ 0:34u2 Þ

0:408ΔA þ γ

ð10Þ

The input variables (A, D, u2 and Ta) in this equation are averaged values on a 24-h scale, starting from hourly scale values measured by standard meteorological stations. –

The second equation allows the calculation on an hourly scale (in millimetre per hour):

ET0 ¼

37 u2 D T a þ 273 Δ þ γ ð1 þ 0:24u2 Þ

0:408ΔA þ γ

ð11Þ

For alfalfa and irrigated grass, the minimum stomatal resistance is always equal to 100 s m−1. Moreover, this resistance is constant during the day and has the same values in different climate (humid, semi-arid or arid). The effective surface for the transpiration, in the case of the reference surface, is only 50 % of the total leaf surface. The leaf surface, for the reference crop, is correlated to the height of vegetation by the following statistical relationship:

For A>0, the input variables (A, D, u2 and Ta) are the hourly values directly measured by standard meteorological stations. The canopy resistance is considered a constant in Eqs. (10) and (11); therefore, the ET0 computed on hourly and daily scales by means of these equations only translates all effects of weather, according to the first hypothesis in the two-step approach.

LAI ¼ 24⋅hc

2.2 The strong points of the FAO-56 methodology

ð8Þ

By taking into account the three previous hypotheses the relationship (7) can be written as: . rc ¼ 100 ð0:5  24  0:12Þ ¼ 70 s m−1 ð9Þ However, in a paper published later that aimed to determine the hourly value of the resistance rc, Allen et al. (2006) abandoned the procedure described by Eqs. (7), (8) and (9). In fact, after a review of the values of rc for irrigated grass, they found in the scientific literature that it varied between 30 and 70 s m−1; hence, the authors recommended a value of rc equal to 50 s m−1 when calculating the reference evapotranspiration ET0 on an hourly scale. The range of rc values found by Allen et al. (2006) appears to be smaller than the values determined by Choisnel et al. (1992). In fact, starting from an experimental study carried out on sites located in the UK, Belgium, France and Italy, the latter authors observed a rc range between 10 and 130 s m−1. Finally, by introducing into Eq. (1) the values of ra and rc retained by Allen et al. (1998) in FAO-56, the two following formulations of ET0 can be written:

The approach proposed by Allen et al. (1998) for determining the reference evapotranspiration ET0 has three advantages: 1. It gives a clear definition of the reference surface evapotranspiration ET0. In fact, ET0 has often been confused with the potential evapotranspiration (PET), including the recent literature (see, for example, Douglas et al. 2009; Zhang et al. 2010; Torres et al. 2011). This last term is equivalent to the maximum possible level of evapotranspiration under given climatic conditions. In reality, PET can be either close to or far from ET0 (see Katerji and Rana 2011 for an extensive analysis), following the type of the evaporative surface (bare soil, water surface, annual crop, forest, etc.). 2. The calculation of ET0 is realised by the PM equation. This equation is based on an analytical description of the energy exchanges between cropped surfaces and atmosphere. Thus, the use of statistical formulations, adopted in the FAO-24 publication and often criticised because they require correction when used in sites submitted to different climates, can be avoided (Smith et al. 1992; Choisnel et al. 1992).

FAO-56 methodology for determining water requirement

3. The determination of ET0, on daily and hourly scales, is only based on input variables (A, D, u, Ta) measurable in standard meteorological stations. Therefore, the proposed method is perfectly operational and suitable for practical purposes. 2.3 The weak points of the FAO-56 methodology The methodology proposed by the FAO-56 publication has several possible sources of error, linked to the simplifying hypotheses introduced by the authors with the aim of making operational the ET0 calculation. The most important sources of error include the following: –

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The adoption of a daily version of the PM equation. This equation is only valid in a permanent regime, i.e. at least on an hourly time scale (see Section 2.1). This choice allowed Allen et al. (1998) to use a 24-h average of weather variables for calculating the ET0. However, the largest amount of water is lost through evapotranspiration only during part of the day, mainly between 10 and 16 h. The use of a 24-h average does not permit faithful estimations, which follow the weather conditions when the evapotranspiration is actually performed. The hourly approach proposed in the following study by Allen et al. (2006) avoids this fault. However, the hourly scale is rarely used to calculate ET0 for practical purposes and almost all studies used the daily version as expressed by Eq. (10). The hypothesis of a constant rc is congruent with the hypothesis that all effects of weather are included in the calculation of ET0. Nevertheless, this hypothesis is in contradiction with the results of the literature that have analysed the variation of stomatal resistance with climatic variables: radiation, air vapour pressure deficit, wind speed and carbon dioxide (see Damour et al. 2010 for an extensive and recent review). In particular, this hypothesis has not been verified in experimental trials carried out in the Mediterranean region on irrigated grass surfaces. In fact, these experiments underlined significant variations in the canopy resistance rc on daily and seasonal scales (Rana et al. 1994; Steduto et al. 2003; Katerji and Rana 2006; Lecina et al. 2003; Perez et al. 2006). Allen et al. (2006) admitted that there were possible variations in the rc of irrigated grass; however, they considered that the error due to neglecting these variations has little impact on the calculation of ET0. The particular constant value of rc, proposed by Allen et al. (1998, 2006), is the result of a partial or total statistical process and is not the result of a true determination procedure. This is clearly the case of the rc value on hourly scale (50 s m−1), which is an arithmetic average deduced by a bibliographic review, as shown

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before. For the rc values on a daily scale, the approach described by Eqs. (7), (8) and (9) is based on three simplifying hypotheses (see Section 2.1.2) which appear to be far from reality when compared with experimental data. Katerji et al. (1983), for example, noted that in the Paris region, during the humid period, the minimum value of the stomatal resistance rs of an alfalfa crop was close to 100 s m−1 only on the upper side of the leaves, although the incident radiation was greater than 400 W m−2. Moreover, on the lower side of the same leaves, this value could be approximately 50 % greater. The range of values of the inclination and stomatal density of leaves can explain these differences (Katerji et al. 1983). Finally, during periods with radiation lower than a threshold value (the earlier and later hours of the day and when clouds are passing), the observed stomatal resistances are two to three greater in both leaves sides. Furthermore, these authors observed that the average experimental values of rs were affected by a standard error between 15 and 25 %, following radiation conditions. Therefore, the estimation of a constant minimum daily stomatal resistance equal to 100 s m−1 is very approximate. Allen et al. (1989) considered that only 50 % of the leaf surface [see Eq. (9)] is actively involved in transpiration (which corresponds to a LAI equal to 1.42 in the case of a reference crop with a height of 0.12 m). Nevertheless, Monteith et al. (1965) clearly demonstrated that in an oat crop, the total leaf surface is active until the LAI is lower or equal to 6. For this reason, the total LAI value was taken into account for the calculation of the actual evapotranspiration of snap beans (Black et al. 1970), soybean and grain sorghum (Brun et al. 1972), corn (Perrier 1976) and alfalfa crops (Katerji and Perrier 1985). Allen et al. (1989) introduced another argument for justifying this coefficient of 50 % when considering the active leaf surface: the use of one half of the total LAI helped to compensate for the use of a 24-h average for weather. In reality, it is a statistical coefficient that accounts for correcting the lack of a permanent regime consideration in the ET0 calculation, as we emphasised before.

These uncertainties affecting the calculation of daily rc are most likely the reasons because Allen et al. (2006) abandoned the procedure when adopting the hourly values of rc. However, these authors did not change the value of 70 s m−1 for rc when calculating ET0 on a daily scale [see Eq (10)]. 2.4 The ET0 calculation following FAO-56: validation in the Mediterranean region The calculation of ET0 on a daily scale [Eq. (10)] as proposed by Allen et al. (1989) has been tested by Steduto et al. (1996)

N. Katerji, G. Rana

and compared with direct measurement, by lysimeters, in five countries of the Mediterranean region: Italy (two sites in the south, Rutigliano and Policoro), Spain, Tunisia, Morocco and Turkey. The test covered almost 3,000 days in total. These authors observed that the ET0 calculated by Eq. (10) underestimated the ET0 measured for the reference grass in this region (see Fig. 1). They confirmed the conclusions obtained by Rana et al. (1994) in a previous study carried out in south Italy. However, these authors were not clear in their explanations of these results. In fact, they considered that the values calculated by Eq. (10) tended to be overestimated at low ET0 values (for example, see the test of Moroccan data) and to be underestimated at high ET0 values. They stated that in the particular case of Turkey, the values calculated by Eq. (10) appeared to strongly underestimate the ET0. Steduto et al. attributed this underestimation to the use of alfalfa as a reference crop for determining the ET0 as well as to advection effects; although they also noticed that the same underestimation was observed using a lysimeter on Turkish grass. The same conclusions were recently reached by Azhar

Fig. 1 Comparison at daily scale between reference evapotranspiration ET0 calculated by the FAO-56 methodology and measured by lysimeter in several sites of Mediterranean region (after Steduto et al. 1996)

and Perera (2011) for the Mediterranean climate of south eastern Australian conditions. In reality, the differences found between the measured and calculated ET0 values were due to the different error sources associated with the FAO-56 methodology, as discussed above. The calculation of ET0 following an alternative solution, as proposed in a next section, will allow us to identify the weight of each error source.

2.5 Alternative solutions for determining ET0 in the Mediterranean region The solutions for correctly determining ET0 can be found by simply overcoming the error sources that affect the method proposed by Allen et al. Summarising, these sources are (1) the lack of the permanent regime and (2) the adoption of a constant canopy resistance rc found with a statistical approach. The first error source can be overcome simply by adopting an hourly scale and the second one can be solved by modelling rc to take into account its variability. Many authors, in the Mediterranean region, proposed specific models for irrigated grass enabling the simulation of canopy resistance rc, on hourly or daily scales, in relation to weather variables (Rana et al. 1994; Todorovic 1999; Steduto et al. 2003; Perez et al. 2006). Other authors (Jarvis 1976; Szeicz and Long 1979; Katerji and Perrier 1983; Hatfield 1985; Shuttleworth 2006; Orgaz et al. 2007) proposed more general models, applicable to irrigated grass as well as to other cropped surfaces (annual, perennial and in-row crops, forests). Some of these ET models are analytical and others are empirical or semi-empirical. Several works were devoted to evaluate and compare the performances of these models in the Mediterranean region (Steduto et al. 2003; Perez et al. 2006; Pauwels and Samson 2006; Shi et al. 2008; Gharsallah 2010; Katerji et al. 2011b) and we address the reader to these references for more details. The ET model proposed by a Katerji and Perrier (1983) is discussed in this study. This model was tested and validated with success for several cropped surfaces. Table 1 presents a review of this works. The Katerji and Perrier model (KP model) enables either the calculation of the ET0 values or the determination of the crop coefficient Kc for a large range of crops and, hence, the actual evapotranspiration. This last point will be addressed specifically in Section 3 of this work. Katerji and Perrier (1983) based the calculation of actual crop ET on the PM approach. However, to determine the resistances ra and rc they proposed a specific procedure. In fact, according to Perrier (1975a, b, c), they proposed calculating the resistance ra between the top of the crop and a reference point (z) located in the boundary layer above the canopy as follows:

FAO-56 methodology for determining water requirement Table 1 Values of the coefficients a and b, for the Katerji and Perrier model (KP), for different crops under well-watered conditions Crop

a

Alfalfa Clementine Forest Grain sorghum

0.31 0.23 0.545 0.54

Grass Grassland on slope Lettuce Oats on slope Soybean Sunflower Sweet sorghum Tomato Vineyard Wheat on slope Corn Canola

0.16 0.42 0.73 0.88 0.55 0.45 0.845 0.54 0.91 0.96 1.497 0.088

ln ra ðzÞ ¼

z−d hc −d ku

b

Reference 0.25 0.0042 1.31 0.61

0 0.51 −0.58 3.39 1.55 0.2 1 2.4 0.45 4.24 −1.718 0.134

Katerji and Perrier (1983) Rana et al. (2005) Shi et al. (2008) Rana et al. (1997a) Rana et al. (1994) Pauwels and Samson (2006) Alves and Pereira (2000) Rana et al. (2011) Rana et al. (1997b) Rana et al. (1997a) Rana et al. (2001) Katerji and Rana (2006) Rana and Katerji (2008) Rana et al. (2011) Liu et al. (2012) Liu et al. (2012)

ð12Þ

where u* is the friction velocity (in metre per second). The wind speed (in metre per second) at the reference point z above the canopy was calculated as follows (Rosenberg et al. 1983; Arya 2001): uð z Þ ¼

u z−d ln z0 k

the resistance (rc) by the following relation (details of the demonstration are given in the Appendix I):

ð13Þ

rc r ¼a þb ra ra

ð15Þ

where a and b are empirical calibration coefficients requiring experimental determination; r* (in second per metre) is written as follows: r ¼

Δ þ γ ρcp D Δγ A

ð16Þ

This resistance r* is linked to the isothermal resistance (ri =ρcpD/γH) introduced for the first time by Monteith (1965), this last variable can be considered as a climatic resistance, because it depends only on weather variables. Moreover, r* represents a critical value for the evaporative process (Daudet and Perrier 1968) because it is a threshold between the situation, rc r*, in which λE decreases with wind speed. By combining, Eqs. (5) and (9) the hourly ETc. (in millimetre per hour) can be written as follows: . ΔA þ ρc D ra p 1    ETc ¼ ð17Þ r λ Δþγ a þbþ1 ra The ET calculation, on a daily time scale, was obtained by summing the hourly values. Thus, on a daily scale the ET can be written as: X ETd ¼ ET ð18Þ h

Therefore, the final formula for ra was written as follows:

ð14Þ

The value of the a and b coefficients, in Eqs. (15) and (17), is specific for each species (see Table 1). For a given species, Katerji and Perrier (1983) identified three situations for when it is necessary to calibrate the a and b coefficients:

The reference point z is normally located 2–3 m above the crop. The experimentally procedure for precisely determining this point z for each crop is detailed in Rana and Katerji (2009). The Eqs. (12) to (14), as well as Eq. (2), are exactly valid only for neutral atmospheric stratification. However, these equations can be adopted in the case of irrigated crop, because the associated discrepancy can be neglected in this condition (Itier and Katerji 1983) Moreover, Katerji and Perrier (1983) considered the canopy resistance rc as variable and not constant. By applying the π-theorem by Buckingham (Kreith and Bohn 2001), they proposed to simulate the variation of

1. Well-watered crops. In these conditions the a and b coefficients calibrated for a given species can be considered as constant and independent on the soil covering for crops with LAI>1 (see theoretical analysis of Katerji and Perrier 1985 and the experimental verifications of Katerji and Rana 2006). Moreover, multi-variety and multi-local tests demonstrated that these coefficients are not sensitive to the variety and to the site where they were determined (Rana et al. 1997b, 2001, 2012). 2. Well-watered crops in the senescence stage (physiological stress). This situation does not concern the irrigated grass. 3. Water-stressed crops during the development stage. The particular procedure followed to calibrate the a and b coefficients in relation to crop water status has been

ln ra ðzÞ ¼

z−d z−d ln z0 hc −d k 2 uz

N. Katerji, G. Rana

described in detail by Rana et al. (1997a; 1997b). It was not considered in the present study which only discusses the irrigated crop. After Eq. (17), the ETc seems to be dependent on two groups of variables that interact simultaneously: 1. The climatic variables A, D, uz and Ta measured at the reference point z above the crop. The first three variables are independent of each other; instead, the variable Ta is strongly linked to the variables A and D; 2. The biological variable of the surface resistance rc. When Eq. (17) is applied on irrigated grass crop to calculate ET0, the determined values do not depend only on the climate (hypothesis admitted in the two-step approach) but also on rc values, which can increase or decrease the effects of climate on ET0. The calibration of the a and b coefficients was performed, in practice, using the hourly data for rc, determined during two or three clear days randomly chosen during the experimental period. The calculation of rc was performed by inverting the PM equation, after introducing in this equation the measured hourly values of ET. Katerji and Rana (2006) demonstrated that approximately 20 values of hourly data were enough to reliably calibrate the a and b coefficients. Figure 2 shows the relationship between the ratio rc/ra and the ratio r */r a at hourly scale for reference grass, observed in Rutigliano site (south Italy) during 3 days in 1990 (after Katerji and Rana 2006). A linear fit resulted in the following values: a=0.16 and b not statistically different

from 0, with r2 =0.58. Table 1 shows a review of the values for the coefficients a and b calibrated on several crops. Therefore, in the case of irrigated grass crop the final expression of the model on hourly scale (in millimetre per hour) is as follows: . 1 ΔA þ ρcp D ra   ET0 ¼ ð19Þ r λ Δ þ γ 0:16 þ 1 ra The calculation of the ET on a daily time scale (in millimetre per day) was obtained by summing the hourly values [see Eq. (18)]. At this point of the analysis we have three methods available to calculate ET0: –

–

–

Method “A,” developed in the FAO-56 publication, which gives ET0 on a daily scale [see Eq. (10)]. This method does not take into account the permanent regime and requires a resistance value that is fixed at a constant value of 70 s m−1; Method “B,” proposed by Allen et al. (2006) for calculating ET0 on an hourly scale [see Eq. (11)], then on a daily scale by summing the hourly values [see Eq. (18)]. This method takes into account the permanent regime, but adopts a constant resistance rc that is fixed equal to 50 s m−1 for irrigated grass; A method “C,” proposed by Katerji and Perrier (1983), which gives ET0 on an hourly scale [see Eq. (19)] then on daily scale by summing the hourly values [see Eq. (18)]. This method takes into account the permanent regime and adopts a variable resistance rc for the irrigated grass.

2.6 Comparison of three approaches to calculate the reference evapotranspiration ET0

Fig. 2 Relationship on hourly scale between the ratio rc/ra and the ratio r*/ra determined for reference grass during May 1990 in Rutigliano site (south Italy) (after Katerji and Rana 2006)

The three methods for calculating ET0, described above, have been tested on hourly and daily scales, by comparison with measurements carried out with a weighing lysimeter in 2003, for 154 days between April and November 2003. The lysimeter, described in detail in a previous publication (Rana et al. 1994), has a surface of 4 m2 with a resolution of 0.06 mm. It is located a few metres near the meteorological station in the reference grass field of 1 ha in the Rutigliano site (south Italy). This site is characterized by a Mediterranean climate; its detailed description during a long period can be found in a previous study (Katerji et al. 2010a). Nevertheless, the same climatic variables A, D, uz and Ta, determined at hourly or daily scales were used in these calculations for the tree methods under study. It should be remembered that summer 2003 was particularly hot, as demonstrated by the meteorological observation carried out in the Rutigliano site during this year (Katerji et al. 2010b), and

FAO-56 methodology for determining water requirement

was considered a meteorological extreme that rarely occurs in Europe (Ciais et al. 2005). The results of this comparison are shown in Figs. 3, 4 and 5. On a daily scale, the ET0 calculated by the method A (see Fig. 3) underestimates the measured values. This result is perfectly comparable with the previous observations presented by Rana et al. (1994) and Steduto et al. (1996) for the same site. However, the observed slope between the calculated and measured values (0.7 in this case) is lower than what was previously observed (0.82), for the same site during 1987–1991 by Rana et al. (1994). This difference could be associated with the exceptional summer conditions of the year 2003. The ET0 calculated by method B, on hourly (Fig. 4a) and daily scales (Fig. 4b), also underestimated the measured values. However, the observed slope on the daily scale (0.85) is much greater than that obtained by method A for the same time scale. Furthermore, the coefficient of correlation, r2, in this case (0.85) is much greater than the one obtained for the method A (0.3). The ET0 calculated by method C, on hourly (Fig. 5a) and daily (Fig. 5b) scales, slightly overestimates the measured values. However, the observed slope on the daily scale is closer to 1 than the one obtained with method B for the same time scale. The coefficient of correlation is also slightly greater than the one obtained with the method A. Finally, the observed slope in 2003 (1.05) is close to the one previously observed (0.98) for the same site by Rana et al. (1994). Therefore, method C appears to be less sensitive to the exceptional conditions of the year 2003 than method A,

a

b

Fig. 4 a Comparison between hourly values of reference evapotranspiration ET0 calculated by the Allen et al. methodology and measured by lysimeter in Rutigliano site (south Italy); b same comparison at daily scale by cumulated of hourly values

Fig. 3 Comparison between daily values of reference evapotranspiration ET0 calculated by the FAO-56 methodology and measured by lysimeter; during April – October 2003 in Rutigliano site (south Italy)

because the climate variability on the canopy resistance is taken into account by this method. After the above analysis and considering the values of the slopes between the calculated and measured ET0 in the three cases [(1) model “A” slope=0.70, (2) model “B” slope=0.85 and (3) model “C” slope=0.98–1.05], the strong underestimation of ET0 as calculated by the FAO-56 methodology can be equally attributed to the lack of the permanent regime and to the neglecting of the variation in rc.

N. Katerji, G. Rana

a

b

Fig. 5 a Comparison between hourly values of reference evapotranspiration ET0 calculated by the Katerji and Perrier (KP) model and measured by lysimeter in Rutigliano site (south Italy); b same comparison at daily scale by cumulated of hourly values

single approach method and only five studies where Kc was determined by the dual approach. Thus, the Kc determined by the single approach prevails for two reasons: (1) the procedure is simple and (2) any fundamental improvement of the accuracy after the application of the dual approach seems to be achieved. Furthermore, after the review of several species by Allen et al. (2005), confirmed by the recent analysis by Odhiambo and Irmak (2012), the observed differences between the values of Kc calculated by the single and dual approaches primarily concern the initial phase of the crop growth cycle. However, the initial phase of a crop cycle is characterised by small leaf surface and particularly low water requirements. For these reasons, the following analysis concerning the determination of Kc in the Mediterranean region will be focused on the single approach. The procedure to determine the Kc by the single approach was described in detail by Allen et al. (1998) in the FAO-56 publication. Only three point values, for Kc, are required to describe and to build the Kc curve; hence three steps are required: 1. (a) To divide the growing period of each crop into three stages that describe the crop phenology or development (initial stage, mid-season and end stages); (b) to determine the lengths of the growth stages for each crop (see Table 11 in the FAO-56 publication) and to identify the three Kc values specific to the crop and corresponding to the three stages previously identified: initial Kcin,, midseason Kcmid and final Kcend stages (see the Table 12 in the FAO-56 publication). However, the generalised KcFAO values of Table 12 are suitable in sub-humid climates, with an average daily minimum relative humidity of approximately 45 % and calm to moderate wind speed averaging 2 m s−1. For other climatic conditions, adjustments are necessary. 2. To adjust the Kc values to the plant height and to climatic conditions (wind speed and relative air humidity averages observed during the relative phenological stage), in case the determination is made under a non-sub-humid climate. The adjustment formulations for the coefficients Kcmid and Kcend are general and applicable to all cultivated crops. They can be written as follows: K cmid;end ¼ K cmid;endðTabÞ

3 The determination of the crop coefficient Kc following FAO-56 The procedure for determining Kc is different in that it can be single or dual. Moreover, it depends on the final objective that can be the irrigation practice or the research purpose. A review devoted to the determination of Kc in the Mediterranean region was performed by Lazzara and Rana (2010). These authors identified 38 studies where Kc was determined by the

 0:3 hc þ ½0:04ðu2 −2Þ−0:004ðRHmin −45Þ 3 ð20Þ where RHmin is the minimum relative humidity in percent. 3. To build a curve, by connecting straight line segments through each of the three growth stages, by adopting the extrapolation procedure for the daily mean values as defined by Allen et al. (1998).

FAO-56 methodology for determining water requirement

3.1 The strong points of the procedure for the determination of Kc The FAO-56 publication gives a guide and tables of Kcini, Kcmid and Kcend values for a large number of different crops, together with the lengths of crop development stages for various planting periods and climatic regions. Moreover, the adjustment of the coefficient Kc in the functions of involving climatic variables is made by using climatic data easily and routinely measured by standard weather stations. Thus, the proposed methodology can be considered as perfectly operational. 3.2 The weak points of the procedure for the determination of Kc Two weak points can be envisaged in the determination of the crop coefficient Kc by the procedure of Allen et al.: –

–

The origin of the Kc values proposed by Allen et al. (1998) in the Table 12 of the FAO-56 publication are not clear (Shuttleworth and Wallace 2009). Allen et al. stated that the values of Kcini were determined in the work of Doorenbos and Kassam (1979), while the Kcmid and Kcend values can be found in Dooenbos and Pruitt (1977), Pruitt (1986), Wright (1981; 1982) and Snyder et al. (1989). However, all these values were determined when the recent formulations of ET0 [Eqs. (10) and (11)] were not proposed, and the methods then used for determining ET0 were justly criticised by Allen et al. (1998). Furthermore, the three values of Kc were not determined in the same experimental conditions as they were attributed to different works and authors. Finally, the Kc values adopted by these authors are calculated as the average of values often resulting in a compromise between very contradictory data, as clearly underlined by Dooenbos and Pruitt (1977), so they should be approached cautiously. The hypothesis for which Kc varies predominately with specific crop characteristics and much less with climate is not confirmed by observations. The reviews of the values of the coefficient Kc observed in different sites (Farahani et al. 2007; Lazzara and Rana 2010) clearly underlined the difficulty of transferring the Kc values into locations, where the local climate and agricultural practises (irrigation method, wetting pattern and frequency and varietal variations) deviate from the condition in which the adjusted values were determined. The possibility of a strong link between the Kc and the local climate would contradict the second hypothesis in the two-step approach.

The ET modelling, presented in Section 2.5, provides a framework in which a theoretical study can be performed on the existing links between the coefficient Kc and the climatic

variables. Here, we propose to carry out this study on tomato crop, one of the most irrigated species in the Mediterranean region. Then, we will compare the results of this theoretical study with the experimental data carried out on the same crop in Foggia site (south Italy) during 3 years (2001, 2002 and 2006). This site is characterised by a Mediterranean climate described in detail over a long period in a previous publication (Katerji et al. 2010a). These data were obtained under the climatic and experimental conditions described in great detail in a recent publication (Rana et al. 2012).

3.3 Theoretical analysis of the Kc variability in relation to the climatic variables that affect crop evapotranspiration The crop coefficient Kc, defined as the ratio between the actual crop evapotranspiration ETc and the reference evapotranspiration ET0, can be explained from the theoretical analysis by the KP model detailed in Section 2.5. By introducing the values of the coefficients a and b, that are specific for tomato crop (from Table 1) into the general Eq. (17), the hourly ET for this crop can be written as: . ΔA þ ρc D t p t ra;t 1   ETc;t ¼ ð21Þ  r λ Δ þ γ 0:54 t þ 2:4 þ 1 ra;t where the subscript “t” indicates the tomato crop. Because the hourly formulation of ET0 is given in Eq. (19), the coefficient Kc on an hourly scale can be obtained from: . ΔAt þ ρcp Dt ra;t   r Δ þ γ 0:54 t þ 2:4 þ 1 ETc;t ra;t . ð22Þ ¼ K c;t ¼ ET0 ΔA þ ρcp D ra   r Δ þ γ 0:16 þ 1 ra This ratio, after some algebraic manipulations, can be written as:

K c;t

γ rt 1þ Δ γ þ Δ ra;t   At γ r Δþγ 0:54 t þ 2:4 þ 1 1þ γþΔ ra;t ¼ γ r 1þ Δ γ þ Δ ra   A γ r Δþγ 0:16 þ 1 1þ γþΔ ra

ð23Þ

By considering that, by the first approximation, the values of the available energy (ΔA/(Δ+γ) of the tomato crop, when

N. Katerji, G. Rana

covering the soil, are very close to that of the irrigated grass ET0, the hourly formulation of Kc for tomato becomes:

3.4 Analysis of the experimental Kc variations on different scales of time and space

  γ rt γ r 0:16 þ 1 a 1þ γ þ Δ ra;t γþΔ ra   ¼ γ r γ rt 1þ 0:54 þ 2:4 þ 1 1þ γ þ Δ ra γþΔ ra;t

Figure 7 shows the following experimental data carried out in a tomato crop cultivated in the Foggia site during two successive days in May 2000:

1þ

K c;t

ð24Þ Figure 6a–c show the behaviour of the coefficient Kc, when determined by Eq. (24) for a tomato crop in the mid-season growth stage (crop height 0.7 m), with respect to the three independent climatic variables A, D and u, which affect the actual evapotranspiration ET [see Eq. (17)]. Consistent Kc changes are observed: (a) with available energy A, for a fixed range of D and wind speed u; (b) with wind speed u, for a fixed range of A and D; and (c) with deficit of saturation D, for a fixed range of A and wind speed u.

a

b

Fig. 6 Hourly values of crop coefficient Kc, for adult tomato crop (height=0.70 m), change, a with available energy A, for range of deficit de saturation D and wind speed u fixed b with wind speed u for range of

– –

The hourly path of the climatic variables A, D and u, affecting rc; The hourly path of Kc, experimentally determined by the ratio “ET measured on tomato crop by eddy covariance technique / ET0 measured on irrigated grass by lysimeter”, together with the mean diurnal values of the coefficient Kc.

The hourly values of Kc varied from half to double in a day; the mean diurnal values of Kc exhibited variations of around +60 % over two successive days. The analysis of the hourly variations of Kc in function of climatic variables is complex because each climatic variable can change in both directions on this scale.

c

available energy A and vapour pressure deficit D fixed, and c with deficit de saturation D for range of available energy A and wind speed u fixed

FAO-56 methodology for determining water requirement

Rana et al. (2012) compared the curve of the coefficient Kc (KcFAO), obtained by the procedure proposed by Allen et al. 1998 (see Section 3) during the growth cycle of a tomato crop in the Foggia site, together with the values of the daily coefficient Kc (Kc,exp), experimentally determined by the ratio “daily ET measured by eddy covariance/daily ET0 calculated by Eq. (10)”. The authors noted that the differences between the daily values of KcFAO and of Kc,exp are evident during the whole growth cycle and, particularly, during mid-season. These differences are about ±40 % of the experimental Kc,exp values with respect the mean values represented by KcFAO; they originate only from the variation in climatic variables, because the managements factors (cited by Farahani et al. 2007 to explain the Kc variations) are the same in the present case. Table 2 presents a review of Kc values found primarily in the Mediterranean climate for the tomato crop. All these studies propose the adjustment of KcFAO values to improve the accuracy of the calculated actual evapotranspiration for this crop. This review claims that the variability between the adjusted values of KcFAO remained in the range of values attributed to the climatic variables. This fact does not exclude a possible effect of management factors on Kc. Nevertheless, here we notice that the effect of climate on the variations in Kc appears to be dominant. The theoretical analysis and the experimental observations lead to concordant conclusions. The presented results

show the inconsistency of the hypothesis that Kc varies little with respect to climate and that those small variations can be taken into account by Eq. (20).

4 The determination of daily ET following the methodology proposed by FAO-56 and alternative solution The daily calculation of the actual ET, proposed by Allen et al. (1998), is typically written as: ETc ¼ K c

900 u2 D T þ 273 Δ þ γ ð1 þ 0:34u2 Þ

0:408ΔA þ γ

ð25Þ

with Kc calculated in this case by following the single approach method. Figure 8a presents, after Katerji and Rana (2006), a comparison carried out on a daily scale for four species (grain and sweet sorghums, soybean and sunflower) cropped in Rutigliano site (south Italy) between the values calculated by FAO-56 methodology (Eq. 25) and those measured by the Bowen ratio method. The observed slopes and intercepts of the linear adjustment varied between 0.38 and 1.24 for slopes and between 0 and 2.61 for intercepts, with the coefficient of correlation, r2, between 0.38 and 0.72. These data were obtained under

Fig. 7 Hourly values of crop coefficient Kc and weather variables (available energy A, deficit de saturation D and wind speed u) observed on adult tomato crop (height, 0.70 m) during two successive days of May 2000 in Foggia site (south Italy)

FAO– 56

Crop

0.19

0.6 1.21

0.8 0.3 0.2 0.2 1 0.2 0.35

var. Brigade

–

cv Ability

cv. Dracula 0.23 cv. Rio Grande 0.83 0.6

1.1 0.8

0.36

0.55

0.65

0.95

0.49 0.54

Late

0.99–1.93 0.6 1.15 0.68 1.15–1.2 0.7–0.9

1.15 1.02 0.96 1.06 1.05 1.05 0.99 1.09 1.08 0.9–1.1

0.81 0.83 0.9–1.15

1.3 1.2

0.77–1.2

Initial Development Middle

Kc

cv. Pull Hybrid PS 1296

Variety

0.76–0.81 0.43–0.68

0.56–0.71

0.8

0.44 0.47 0.7

0.8 0.9

0.74

End

0.7

Season

1 1

IV 2

III

II

2 I

2

2

2

– Sprinkler Semi-humid

Drip with mulch Drip with mulch Furrow Furrow Drip Furrow Drip Drip (H2003) Drip (D2003) Furrow Drip Drip

Drip

Drip

Number Irrigation of years method

Ebro Valley

Jordan Valley

Bechar, Tamanrasset Puglia

Reference

Amayreh and Al-Abed (2005) Mediterranean Vazquez et al. (2005)

Semi-arid

Mediterranean Rinaldi and Rana (2004)

Mediterranean Hamidat et al. (2002)

Climate

Italy Tunisia

Italy

Policoro –

Basilicata

Mediterranean Čereković et al. (2010) Mediterranean Yacoubi et al. (2010)

Mediterranean Favati et al. (2009)

Chile Maule region Ortega-Farias et al. (2006) California San Joaquin Valley Mediterranean Hanson and May (2005)

Spain

Jordan

Italy

Algeria

Country

Table 2 Review of Kc values determined during different crop growth stages for the tomato crops in Mediterranean or semi-arid climates. In addition, the values recommended by the FAO-56 publication

N. Katerji, G. Rana

FAO-56 methodology for determining water requirement

climatic and experimental conditions described in great detail in Katerji and Rana (2006). A similar comparison, after Rana et al. (2012), was performed on a tomato cropped in the Foggia site during 3 years (2000, 2001 and 2006), between daily ET values measured by the eddy covariance method and those calculated by Eq. (25). This comparison permitted a quantification of the inter-annual variability that affects the parameters (slope, intercept and correlation coefficient r2) of the linear adjustment determined between measured and calculated values. The observed values of these parameters (see Fig. 9a–c) varied between 0.80 and 0.88 for the slope and between 0.31 and 1.4 for the intercept. Furthermore, the coefficient of correlation remained quite stable (0.53–0.55) during the three years of the experiment. The weakness of the estimation of the ET calculated by applying the FAO-56 methodology is the result of the accumulation of errors that affects the calculation of either ET0 or Kc. When the term ET0 in Eq. (25) is measured by a lysimeter rather than calculated by Eq. (10) the accuracy of the estimation is improved, because the determination of Kc becomes the only source of error. Moreover, in previous study, Rana and Katerji (2008) compared, on overhead vineyard crop during two years 1996 and 1997 in Rutigliano site, the daily ET measured by the Bowen ratio method and calculated by Eq. (25), in which the ET0 values were measured by lysimeter. The observed values of slope (0.83), intercept (∼0) and r2 (0.76) in this study were better than those previously observed on other crops, as shown in Figs. 8a and 9a–c. The one-step approach has been explained in detail in Section 2.5. It calculates the ET of a crop directly, without involving a reference surface ET0. This approach is based on fewer computation steps and, consequently, on fewer error sources than the two-step model. An example of the improvement in the estimation of ET by the one-step approach is presented in Fig. 8b (after Katerji and Rana 2006). It was carried out by applying the model KP at daily scale, using the Eqs. (17) and (18), on the same site, the same crops and the same experimental data, previously presented in Fig. 8a (i.e. grain and sweet sorghums, soybean and sunflower). The parameters of the linear adjustment between the measured and calculated values indicated a weak variation of the slope (between 0.94 and 1.01) and a strong stability of the intercept (systematically close to 0), with r2 ranging between 0.68 and 0.94. The comparison of observed parameters of linear adjustment for each crop, shown in Fig. 8a, b, allows the quantification of the improvement obtained by adopting the one-step approach instead of the two-step approach. The multi-annual test for the KP model was carried out during 3 years on the same site, the same tomato crop and

a

b

Fig. 8 Comparison, for four crops, between daily evapotranspiration measured in Rutigliano site (south Italy) by Bowen ratio method and calculated by the FAO-56 methodology (a) and by Katerji–Perrier model (b) (after Katerji and Rana 2006)

the same experimental data previously presented in Fig. 9a–c. The use of the KP model decreases the interannual variability, which affects the linear adjustment parameters previously observed in Fig. 9a–c. In fact, the determined slopes during three years (Fig. 9d–f) varied only between 0.95 and 0.99, while the intercepts remained systematically equal to 0. Furthermore, the coefficient r2 varied between 0.76 and 0.93, but it remained greater than what was previously determined (0.53–0.55) when ET was calculated by FAO-56 methodology (Eq. 25).

N. Katerji, G. Rana

a

d

g

b

e

h

c

f

i

Fig. 9 Comparison for tomato crops during 2000, 2001 and 2006 between daily actual evapotranspiration measured in Foggia site (south Italy) by the eddy correlation method and calculated from the FAO-56

methodology (a, b and c), calculated from the original (d, e and f) and operational version (g, h and i) of Katerji–Perrier (KP) model

5 Calculation of crop ET by an operational one-step approach

Recently, Rana and Katerji (2009) developed an operational version of one-step model to calculate ET, using only the variables collected routinely by standard weather stations. In the operational version, the net radiation is not measured but calculated by a linear relation as follows:

In the one-step approach, the ET determination by Eq. (17) requires specific measurements of weather variables at the reference point z above the canopy (see Section 2.5). These measurements are usually not routinely acquired. For this reason, the one-step approach is not considered to be operational and this is the primary obstacle concerning the use of this approach in practice (Farahani et al. 2007).

0

Rn ¼ a Rg þ b

0

ð26Þ

For the calculation of the available energy (A), the soil heat flux (G) was considered as a constant equal to 0.1 Rn.

FAO-56 methodology for determining water requirement

The values of the a′ and b′ coefficients [see Eq. (26)] were experimentally calculated, after a regression between the net radiation (Rn) values measured above the crop and global radiation (Rg) values measured at the standard weather station. The values of the a′ and b′ coefficients were similar to the values determined for other crops by other authors (see a synthesis in Table 3). Moreover, also the two climatic variables Ta and D are not directly measured above the irrigated crops, but considered similar to values measured at standard weather stations (see demonstration in Rana and Katerji 2009). According to theoretical analysis of Rana and Katerji (2009), the wind speed at the top of the crop boundary layer can be written as follows:     u z−d ucrop ðzÞ ¼ ln ð27Þ z0 crop k grass In this equation, the friction wind speed (u*) is determined for a grass, at a standard weather station, by the following relation:   u ustation  ¼  ð28Þ zsensor −d grass k grass ln z0;grass where dgrass and z0,grass are linear functions of the height (h) of the grass crop (d=0.75h and z0 =0.1h); and ustation is the wind speed measured at height zsensor, which is usually set at 2 m above the soil surface. In summary: all the climatic variables (Rg, T, D and u) in the operational version of one-step model are currently measurable by standard stations. This calculation needs only the height of the crop. Therefore, the operational version of the model can be applied routinely and can be easily made automatic.

The robustness of the operational version has been validated during the cycle of life of soybean, sweet sorghum (Rana and Katerji 2009) and tomato crops (Rana et al. 2012). Figures 9g–i present, as an example, a comparison on daily scale carried out during three years on tomato crop (on the same site, on the some crops and with the same data previously presented in Fig. 9a–f) between the ET values measured by the eddy correlation method and calculated by the operational version of the KP model. In the case of the operational version, inter-annual variations of the slope (between 1.03 and 1.06) and intercept (always equal to 0) were weak. Only variations of r2 (between 0.69 and 0.96) were noticed. The procedure to change from the original version of onestep model to the operational version weakly affects the quality of the ET estimations. In fact, the parameters of the linear adjustment (slope, intercept and correlation coefficient r2) determined between the measured and calculated by the two versions are close. Thus, the primary obstacle (Farahani et al. 2007) concerning the use of the one-step model in practice appears to be overcome.

6 Synthesis and conclusions The FAO-56 publication can be thought as an attempt to improve the determination of actual crop evapotranspiration following the critical observations after an expert consultation (Smith et al. 1992). The correct evaluation of crop water requirements is a key component of any strategy to improve agricultural water management especially in the Mediterranean region, where water resources are particularly scarce. The debate around the improvement of the determination of ET values was primarily focused on two options: –

Table 3 Coefficients a′ and b′ for determining net radiation (Rn), from global radiation (Rg), using Eq. (26) Crop

a′

b′ (W m−2)

References

Wheat Barley Durum wheat Oats

0.751 0.665 0.808 0.75

−76.8 −83.8 −121.5 −116.6

Fritschen (1967) Fritschen (1967) Fritschen (1967) Fritschen (1967)

Cotton Grain sorghum Maize Rice Sunflower Soy bean Sweet sorghum Tomato

0.738 0.752 0.82 0.865 0.76 0.79 0.56 0.74

−69.8 −74.0 −90.5 −90.5 −41.1 −109.0 −22.2 −44.5

Fritschen (1967) Fritschen (1967) Uchijima (1976) Uchijima (1976) Abi Saab (2007) Rana and Katerji (2009) Rana and Katerji (2009) Rana et al. (2012)

–

The adoption of the two-step approach “reference evapotranspiration (ET0)—crop coefficient (Kc)” by introducing an improvement of the ET0 calculation by means of the PM equation. The adoption of the one-step approach with the aim to reduce the errors associated with the intermediate calculation of the two-step approach.

By adopting the former in the FAO-56 methodology, Allen et al. (1998) proposed the following two hypotheses to justify this approach (Allen et al. 2005): – –

The daily reference evapotranspiration ET0, determined by Eq. (10), takes into account all effects of weather; The Kc coefficient varies predominately with specific crop characteristics and only slightly with climate, and this enables the transfer of Kc standard values between locations and climates.

N. Katerji, G. Rana

The analysis presented in this study showed that the first hypothesis is not correct for two reasons: (1) the Eq. (10), adapted by the PM equation, uses climatic variables determined on a 24-h average scale. However, the PM formulation is only valid in a permanent regime, in other words, at least on an hourly scale; (2) formulas (10) and (11) attributed a constant value of the canopy resistance rc to the irrigated grass, in order to meet the requirements of the hypothesis for which ET0 essentially represents all effects of weather. However, the canopy resistance is variable and can have values far from those adopted by Allen et al. The two previous reasons are error sources that affect the accuracy of the ET0 calculation in the Mediterranean region. These errors affect about ±20 % of the differences between the measured and calculated daily ET0. Furthermore, the errors can reach ±30 % of the calculated value during exceptional years, as was the case in 2003. Following our analysis the observed errors can be attributed equally to the lack of a permanent regime and to the neglecting of the variation in rc. Furthermore, also the second hypothesis in the two-step approach is not correct. Starting from a theoretical analysis and from experimental trials carried out on different time and space scales, it was demonstrated that the values of Kc varied greatly in relation to the climatic variables (A, D and u), contrary to the hypothesis stated by Allen et al. (1998). This does not support the possibility of the transferability of the Kc values into locations where the local conditions deviate from the conditions in which the adjusted values of Kc were determined. Finally, the origin of the Kc values proposed by Allen et al. (1998) was not correctly identified, which represented a further source of error. Therefore, the weakness of the ET estimation by the Allen et al. (1998) demonstrated in the present work appears to be foreseeable because this estimation compounds the previously identified errors in the calculation of both ET0 and Kc. The one-step approach could be considered to be a good opportunity to improve the calculation of ET0. This has been demonstrated by several authors in studies carried out in the Mediterranean region (i.a. Testi et al. 2004; Katerji and Rana 2006; Orgaz et al. 2007; Lovelli et al. 2008) as well as in other sites (i.e. Irmak and Mutiibwa 2009). The improvement in the estimation of ET by this approach was demonstrated for several species, annual and perennial, by multiannual tests. Operational versions of the one-step approach were built and validated (Rana and Katerji 2009; Rana et al. 2012). They permitted an estimation of ET using climatic variables collected routinely by weather stations and can be considered to be a suitable and promising method for the future. Shuttleworth and Wallace (2009) considered that the poor performance of the FAO-56 methodology in a semi-arid region can be predicted because the humid conditions are

an implicit prerequisite in this methodology, which rarely occurs in this region. The analysis carried out in the present study supports this issue by solid arguments.

Appendix I The Eq. (1) can be written in the following form: . A þ ρcp D ðra ΔÞ  .  .  λE ¼ 1 þ γ Δ 1 þ rc ra

ðIaÞ

thus the actual evapotranspiration can we considered as function of several variables as:  .  ðIbÞ λE ¼ f rc ; A; D; ra ; ρcp ; 1 Δ The dimensional analysis based on the π-theorem by Buckingham stated that a function depending on n variables of m dimension can be studied by means of n–m dimensionless groups; so the relationship between the canopy resistance rc and the other variables can be written as:  .  ðIcÞ g rc ; A; D; ra ; ρcp ; 1 Δ ¼ 0 Since in the present case the involved units are 4 (mass M, length L, time t and temperature T) and the number of variable is 6, we can search for a relationship involving 6−4=2 groups of dimensionless combination of variables to describe the canopy resistance. Katerji and Perrier (1983) choose the following four main variables (among the 6 above mentioned) to establish the two dimensionless groups: A D ra ρcp

Mt −1 M L−1 t tL−1 −1 −1 −2 ML T t

These variables can be combined with the ones in the Eq. (Ic) in order to give two dimensionless groups and to give the value 1 for the other variables as:   ρcp D rc g ; 1; 1; 1; 1; ¼0 ðIdÞ Δra A ra By taking into account the definition of the critical resistance r* [see Eq. (16)], the above relationship can be written under the form: 

rc γ r g ; 1; 1; 1; 1; Δ þ γ ra ra

 ¼0

ðIeÞ

FAO-56 methodology for determining water requirement

rc ra

This means that we can search for a relationship between γ r γ and γþΔ ra or, since the term γþΔ is almost constant and 

dimensionless, between rrac and rra , which was found (Katerji and Perrier 1983) to be linear under the form rc r ¼a þb ra ra

ðIf Þ

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