Water Resources Management

, Volume 32, Issue 15, pp 5007–5026 | Cite as

Canopy Resistance and Actual Evapotranspiration over an Olive Orchard

  • Athanasios Margonis
  • Georgia PapaioannouEmail author
  • Petros Kerkides
  • Gianna Kitsara
  • George Bourazanis


Τhis study evaluates the hourly actual evapotranspiration (AΕT), predicted either by the two modified Penman-Monteith models (PM) which take into account the canopy resistance (rc) from the Katerji-Perrier (KP) or Todorovic (TD) models, or the simplified PM model with zero rc, as proposed by Priestley and Taylor (PT). The evaluation is based on comparisons with experimental measurements of AΕT applying the ‘Bowen ratio’ method. Hourly experimental data, of air temperature, humidity, wind speed and radiation balance measurements, taken at a 0.5 ha olive orchard in the rural area of Sparta (37° 04΄ N, 22°05΄ E), during the period from June 2010 up to July 2014, are used. The rc estimated by KP model is parameterized by a semi-empirical approach which requires a simple calibration procedure, while rc from TD model is parameterized using a theoretical approach. For estimating AET from minimum data (air temperature, humidity and radiation balance components) the PT model is also employed, since rc is not required and the aerodynamic term of PM is taken into account in the empirical parameter of the model. The results show that PT and KP models are the most appropriate [Refined Index of Agreement (RIA) equal to 0.89 or 0.88, respectively] followed by the TD model (RIA = 0.78). PT or KP models underestimate AET by 9.3% or 9.8%, respectively, while TD model overestimates AET by 15.0%, increased up to 25.8%, during warm period.


Actual evapotranspiration Canopy resistance Penman-Monteith method Katerji-Perrier method Todorovic method Priestley and Taylor method 

1 Introduction

The accurate evaluation of actual evapotranspiration (AET) at different scales is considered as a subject of high interest for hydrology, for determining the water requirement in irrigated areas and analyzing the impacts of climate change on water resources. AET is a component of the water balance of the considered surface and its energy equivalent is the latent heat flux (λΕ) (Penman 1948). Several techniques provide direct measurements of λΕ, such as the Bowen ratio energy balance method (BREB method) or the eddy covariance system. The BREB method has been used to quantify λΕ over various surfaces, such as open water (Burba et al. 1999), grassland (Pauwels and Samson 2006), crop land (Todd et al. 2000; Zhang et al. 2008) and forest (Dawson 1996). Furthermore, λΕ is estimated, using empirical or mechanistic models, based upon in situ observed meteorological variables. Among them, the Penman–Monteith combination equation (Monteith 1965) is commonly used. The model is based upon the assumption that the water has to diffuse firstly through leaves against a canopy resistance (rc), before diffusing into the atmosphere against an aerodynamic resistance (ra), in order to have all λΕ accessible by the plant canopy. According to several researchers, the sensible heat flux (H) only has to diffuse into the atmosphere against the aerodynamic resistance (Shuttleworth 1993; Pauwels and Samson 2006). Canopy resistance expressed as a function of external conditions is usually difficult to be calculated accurately. Thus, a semi-empirical approach has been suggested by Katerji and Perrier (1983), with rc depending on climate variables and the aerodynamic resistance, and requiring in situ calibration of its parameters (KP model). A mechanistic model (not requiring in situ calibration) for determining λΕ was also developed by Todorovic (1999) with the rc being also a function of climatic variables and aerodynamic resistance (TD model). The Priestley-Taylor (PT) model (Priestley and Taylor 1972) has been also used for estimating λΕ (Flint and Childs 1991), as a simplified form of PM method. The PT method has been reported as successfully simulating λΕ over forest ecosystems (Fisher et al. 2005; Kumagai et al. 2005), crop fields (Utset et al. 2004; Liu and Lin 2005) and grassland (Sumner and Jacobs 2005; Pauwels and Samson 2006). The model has been quite often used for calculating potential evapotranspiration. Priestley and Taylor (analysing data over water surfaces and saturated land areas after rainfall) recommended the value of 1.26 as the most appropriate for parameter α (of the respective equation) and thereafter, this value has been used in many studies referring to wet or irrigated crops and grass (Stewart and Rouse 1977; Liu and Lin 2005). When the method is used for estimating λΕ, values of α have been reported as equal to 1.18 over temperate forest during the development period (Shi et al. 2008) or 0.83 or 0.74 (values different for each year of the measurements) above alpine grass (Zhu et al. 2014). Pauwels and Samson (2006) reported an average daily value of α equal to 1.21 for the whole year (with great annual fluctuation) over a natural meadow. They also found α as decreasing with the increase of the water vapour deficit and as increasing with the increase of the soil moisture. Gavin and Agnew (2004) found the value of α equal to 1.25 for periods characterized by floods and equal to 0.80 for periods without surface water, when working over wet meadow during the period March–September. In addition, they concluded that the use of a single value for α throughout the year leads to an underestimation of the λΕ under wet ground conditions and overestimation under dry conditions.

The KP model has been tested successfully by several researchers recommending it as a fairly reliable model for a large number of natural types of vegetation or crops: alfalfa (Katerji and Perrier 1983), grass (Rana et al. 2001), lettuce (Alves and Pereira 2000), tomato (Rana et al. 2012), sweet sorghum (Rana et al. 2001; Katerji et al. 2011), rice (Peterschmitt and Perrier 1991), sunflower (Rana et al. 1997a), soya bean (Rana et al. 1997b; Katerji et al. 2011), wheat (Amazirh et al. 2017), maize and vineyard (Li et al. 2015), overhead table grape vineyard (Rana and Katerji 2008), vineyards in rows (Li et al. 2009), clementine orchard (Rana et al. 2005), orange orchard (Ayyoub et al. 2017), forest (Shi et al. 2008). However, in situ calibration is required to determine the parameters of the KP model. The TD model has been reported (Katerji et al. 2011), as inadequate in correctly assessing λΕ over natural vegetation (prairie, forests) and annual crops (soya bean, sweet sorghum) or perennial crops (vines), as it leads to overestimation of the order of 30–50%, but instead it underestimates λΕ above irrigated grass, by about 20%. It must be noticed that the TD model is an attractive method for parameterization of rc and estimation of λΕ, since it is a theoretical model that does not require the use of additional experimental data for its calibration.

The main objective of this study is to assess the evaluation of accurate hourly λΕ predictions, obtained either from the two modified PM models, with rc estimated by the KP or TD models or the simplified model given by the PT method with zero rc. The evaluation is based on comparisons with measurements of λΕ, taken over olive trees during the period June 2010 to July 2014. Furthermore, the discrepancies of the diurnal or annual variation of all estimated rc and λΕ from the corresponding variations of the experimental rc and λΕ are investigated. Additionally, the effect of rc parameterization factors [water vapor deficit (D), wind speed (U), net radiation (Rn) and water status] on estimating λΕ by KP and TD model and the influence of D and soil moisture on the PT parameter are also examined.

2 Data and Methods

2.1 Data

In this study, the experimental field, covering an area of 0.5 ha (50 m × 100 m) consisted of 132 olive trees (cv. Koroneiki) 15 years old, planted at 6 × 6 m distances is located at the rural area of Sparta (lat. 37° 04΄ N, log. 22°05΄ E, altitude 0.212 km). Ten minutes meteorological- and radiation- data recorded over the olive trees, during the years 2010, 2011, 2012 and 2014 are used. Especially, air temperature, relative humidity, wind speed and wind direction were recorded at 6 and 10 m height above ground and sunshine duration, upward and downward short- and long wave- radiation were recorded at 6 m above ground. Fetch requirements as referred by Perez et al. (1999) were satisfied in the wider site. Campbell instruments were used for measuring air temperature and humidity (MP100 Rotronic temperature/humidity sensors), wind velocity (A 100 R anemometers) and wind direction (W200 P sensor). The short wave- or the long wave- radiations were measured by Eppley Precision Spectral Pyranometers or Eppley Precision Infrared Radiometers, respectively. The sunshine duration was recorded by a Kipp and Zonen CSD3 sensor. Additionally, hourly volumetric soil moisture content measurements taken at 10, 20, 30, 40, 60, 80 and 100 cm depth by using PR2/6 soil moisture profile probe and HH2 Moisture meter (Delta-T Devices, 2013), during the same period were also used. The olive trees were irrigated every 7–10 days, during the summer of 2011 and 2012, in agreement with the local practice to irrigating olive trees every 6–15 days. A pressurized irrigation pipe network was installed in the field, with two sprinklers per tree, providing 90 L/h. The duration of each irrigation event was approximately 2 h. Flow meters were installed in order to monitor the uniformity of the irrigation water implementation. The measurements verified that the irrigation depths were approximately 1 m, which is the effective root depth of olive tree [maximum rooting depth for olive tree is 1.2–1.7 m and 65% of this is the effective root depth (Allen et al. 1998)]. The soil characteristics for three depths of the experimental field are presented in Table 1.
Table 1

Soil characteristics

Depth (cm)





7.19 ± 0.25

7.5 ± 0.17

7.66 ± 0.27


(μS /cm)

1963 ± 230

1677 ± 258

1683 ± 196

Particle size class




Organic matter


6.3 ± 0.94

3.7 ± 0.28

2.5 ± 0.27

N (Kjeldahl)

0.2 ± 0.015

0.1 ± 0.064

0.07 ± 0.02


0.51 ± 0.31

1.02 ± 1.45

2.02 ± 2.53

SCL: sandy-clay-loam, C: clay, CL: clay-loam

The data belonging to days with rainfall or irrigation events were removed from the further analysis.

2.2 Methods

2.2.1 Measurement of Hourly Actual Evapotranspiration

Hourly measured λΕ was obtained from the Bowen ratio/energy balance (BREB) method by the equation:
$$ \lambda E=\frac{A}{1+\beta } $$
where, λΕ is the latent heat flux (W/m2), A is the available energy (W/m2) equal to Rn-G,where Rn is the net radiation (W/m2) and G the ground heat flux (W/m2) and β is the Bowen ratio. The Bowen ratio represents the ratio of the sensible heat flux H (W/m2) over λE and may be expressed by the finite differences of the air temperatures T2 and T1 (°C) and the actual water vapor pressures of the air e2 and e1 (kPa), measured at two heights z2 and z1 (m) respectively, as follows (Bowen 1926; Tanner et al. 1987; Allen et al. 2011):
$$ \beta =\frac{H}{\lambda E}=\frac{\rho {c}_p\left[{T}_2-{T}_1+\varGamma \left({z}_2-{z}_1\right)\right]}{\left(\rho {c}_p/\gamma \right)\left({e}_2-{e}_1\right)} $$
where ρ is the air density (kg/m3), cp is the specific heat of moist air (J/kg0C), Γ is the dry adiabatic lapse rate (0C/m) and γ is the psychrometric constant (kPa/0C).

Hourly values of all parameters were taken into account, after applying criteria for appropriate β values according to Ohmura (1982) and Tanner et al. (1987).

2.2.2 Estimation of Hourly Actual Evapotranspiration

Hourly values of λΕ were calculated by the Penman-Monteith method (Allen et al. 1998) as:
$$ \lambda E=\frac{\varDelta A+\rho {c}_pD/{r}_a}{\varDelta +\gamma \left(1+\frac{r_c}{r_a}\right)} $$
where Δ is the slope of the saturation vapor pressure versus temperature function (kPa/0C), D is the vapor pressure deficit of the air (kPa), ra is the aerodynamic resistance (s/m) and rc is the bulk canopy resistance (s/m).

In order to estimate hourly rc, needed in the PM method, the Katerji-Perrier (Katerji and Perrier, 1983) and the Todorovic (Todorovic, 1999) models were applied over olive trees. For estimating λΕ without requiring rc, the PT model was employed and the aerodynamic term of PM was taken into account in the dimensionless empirical parameter α of the model.

Katerji and Perrier (1983) suggested simulating rc in eq. 3 [after establishing a linear relationship between the two ratios rc/ra and r*/ra, where r* is a climatic resistance (Monteith1965)] by the following relation:
$$ \frac{r_c}{r_a}=a\frac{r^{\ast }}{r_a}+b $$
Where α and b are empirical coefficients requiring experimental determination and r* is defined as:
$$ {r}^{\ast }=\frac{\varDelta +\gamma }{\varDelta \gamma}\frac{\rho {c}_pD}{A} $$
By combining eqs. 3, 4 and 5, the hourly λΕ can be written as:
$$ \lambda E=\frac{1+\frac{\gamma }{\gamma +\varDelta}\frac{r^{\ast }}{r_a}}{1+\frac{\gamma }{\gamma +\varDelta}\left(\alpha \frac{r^{\ast }}{r_a}+b\right)}\frac{\varDelta }{\varDelta +\gamma }A $$
The aerodynamic resistance ra is calculated between the top of the crop hc and a reference plane z within the boundary layer over the crop (Perrier 1975):
$$ {r}_a=\frac{\mathit{\ln}\frac{z-d}{z_0}\mathit{\ln}\frac{z-d}{h_c-d}}{k^2{u}_z} $$
where hc is the mean height of the crop (m), d is the zero plane displacement (m), k is the von Karman constant, uz (m/s) is the wind speed at the reference point z and z0 is the roughness length (m). The application of this approach has the advantage of taking into account Rn, D and ra (the set of climatic variables affecting rc), but is limited requiring specific (based on experimental data) calibration of the coefficients a and b in eq. 4.
In the parameterization proposed by Todorovic for the λE modeling, rα is estimated according to Thom (1972, 1975) between the levels d + z0 for momentum and d = z0h for heat and latent heat fluxes and a reference point z located in the boundary layer above the canopy, by the relation:
$$ {r}_a=\frac{\mathit{\ln}\frac{z-d}{z_0}\mathit{\ln}\frac{z-d}{z_{0h}}}{k^2{u}_z} $$
where z0h is the roughness length to the heat flux (m) equal to 0.1z0.

The difference between the two expressions of the aerodynamic resistances (Eqs. 7 and 8) is the additional aerodynamic resistance introduced by Perrier (1975) for taking into account the structure of vegetation, since he considered that the aerodynamic model of Thom (1972, 1975) ignores the role of plant architecture.

According to Todorovic (1999) and Steduto et al. (2003), “canopy resistance rc causes the difference between the latent heat flux of potential evaporation (λEp) obtained when rc=0 with latent heat flux of actual evapotranspiration (λE), where rc>0, that is λE < λEp. The increase of rc causes a decrease of latent heat flux and an increase in sensible heat flux because the input of available energy (A=Rn-G) remains the same for the system. The additional sensible heat flux causes a higher canopy temperature which corresponds to temperature difference t”. In TD model (Todorovic, 1999) rc was calculated from a quadratic equation as follows:
$$ {a}^{\ast }{X}^2+{b}^{\ast }X+{c}^{\ast }=0 $$
where \( {a}^{\ast }=\frac{\varDelta +\gamma Y}{\varDelta +\gamma } YD \)b =  − γΥt, c =  − (Δ + γ)t in KPa
$$ X=\frac{r_c}{r_i},Y=\frac{r_i}{r_a} $$

\( {r}_i=\frac{\rho {c}_pD}{\gamma A} \) is defined as a climatological resistance (s/m).

\( t=\frac{\gamma D}{\varDelta \left(\varDelta +\gamma \right)}\kern0.5em \)is the temperature difference in 0C.

Canopy resistance rc can be calculated from only one positive solution of the quadratic eq. 9.

Since some micrometeorological variables needed for the PM model are frequently unavailable, a simplified version with rc = 0, was proposed by Priestley and Taylor (1972) in order to estimate λΕ as:
$$ \lambda E=\alpha \frac{\varDelta }{\varDelta +\gamma }A $$
where α is the dimensionless Priestley–Taylor parameter. In this study, when PT model was employed for estimating λΕ, calibrated values of α were used, as obtained by inverting eq. 10 and using the calibration set of data.

2.2.3 Evaluation of Estimated Hourly Actual Evapotranspiration

All the obtained estimates of λΕ were compared with the corresponding measured ones. The comparisons were evaluated by the results of linear regressions [determination coefficient (R2) and slope (a)] and “difference measures” [root mean square error (RMSE), mean bias error (MBE), mean absolute error (MAE) and the new refined index of agreement (RIA)]. RIA introduced by Willmott and Matsuura (2005) is dimensionless, bounded by −1.0 and 1.0 and is a reformulation of Willmott’s index of agreement (IA), which was developed in the 1980s. It is in general, more rationally related to model accuracy than are other existing indices’ (Willmott et al. 2012). The “difference measures” are determined as:
$$ MAE=\frac{1}{n}{\sum}_{i=1}^n\left|{\mathrm{P}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right| $$
$$ \mathrm{MBE}=\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}\mid {\mathrm{P}}_{\mathrm{i}}-{O}_{\mathrm{i}}\mid $$
$$ \mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\left|{P}_{\mathrm{i}}-{O}_{\mathrm{i}}\right|}^2} $$
$$ {\displaystyle \begin{array}{l} RIA=1-\frac{\sum_{i=1}^n\mid {P}_{\mathrm{i}}-{O}_{\mathrm{i}}\mid }{2{\sum}_{i=1}^n\mid {\mathrm{O}}_{\mathrm{i}}-\overline{O}\mid}\mathrm{When}:{\sum}_{i=1}^n\mid {\mathrm{P}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\mid \le 2{\sum}_{i=1}^n\mid {O}_{\mathrm{i}}-\overline{O}\mid \mathrm{or}\\ {} RIA=\frac{2{\sum}_{i=1}^n\mid {\mathrm{O}}_{\mathrm{i}}-\overline{O}\mid }{\sum_{i=1}^n\mid {P}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\mid }-1\mathrm{When}:{\sum}_{i=1}^n\mid {P}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\mid >2{\sum}_{i=1}^n\mid {\mathrm{O}}_{\mathrm{i}}-\overline{\mathrm{O}}\mid \end{array}} $$
where, Pi is the estimated value by the model and Oi the observed one.

In this study the percentage MBEs and RMSEs are estimated.

3 Results

3.1 Estimation of Hourly Canopy Resistance and Actual Evapotranspiration

3.1.1 Calibration of KP and PT Model

Twenty values of hourly data have been referred as enough for reliable calibration of the KP model (Katerji and Rana 2006). This relatively small number of observations does not seem a strong constraint for the model applicability. In this study, the KP model was calibrated for cold and warm period separately, using hourly values of measured rc (rcM), determined during four clear days of cold period and three clear days of warm period respectively. These days were randomly chosen during the year 2011. The period from October to April is defined as cold period, while the period from May to September, characterized as cultivation period for olive trees [according to Michelakis et al. (1996)] is defined as warm period. The rcM was calculated by introducing the hourly measured λE (λEM) and the ra, employed by the KP model (raKP), in eq. 3. The calibration yields the coefficients a and b from the relation rcM/raKP as a function of r*/raKP (eq. 4). Figure 1 represents the results of the linear regression between rcM /raKP and r*/raKP, during clear days of cold period (a) and clear days of warm period (b). The values of the slope, intercept and R2 obtained by the linear regression analysis are 0.69, 2.43 and 0.91 respectively, for the cold period and 0.94, 1.37 and 0.99 respectively, for the warm period.
Fig. 1

Results of the linear regression between the ratio of the measured canopy resistance over the KP aerodynamic resistance (rcM/raKP) and the ratio of the climatic over the KP aerodynamic resistance (r*/raKP) for clear days of cold period (a) and clear days of warm period (b)

Using only one value for parameter α, throughout the year, the estimated λE by PT method was found as overestimating λEM, during the warm period and underestimating it during the cold period. So, the calibration of the PT model was also carried out for cold and warm period separately, using the same hourly calibration data set. The values of the parameter α were determined by the slope obtained by linear regression analysis between λEM and Δ (Rn-G)/ (Δ + γ) (eq. 10). The results, presented in Fig. 2, show values of α equal to 1.07 or 1.01 for the cold (a) or the warm (b) period, respectively.
Fig. 2

Results of the linear regression between hourly measured λΕ (λΕM) and the ratio Δ(Rn-G)/(Δ + γ) for clear days of cold period (a) and clear days of warm period (b)

3.1.2 Evaluation of Estimated Hourly Actual Evapotranspiration

The linear regressions between estimates of hourly λΕ by PM equation, using rc from the TD- (λETD) and KP- (λEKP) method or PT (λEPT) method and hourly λEM are presented in Fig. 3 (a, b and c, respectively) considering all the data (except the calibration set of data). Results of the linear regressions (a and R2) and ‘difference measures’ (RMSE, MBE, RIA and MAE) between estimated by TD or KP or PT models and λEM for all or warm period data are presented in Table 2. It is apparent that the PT model simulates λΕM very well (R2 = 0.99, RMSE = 14.88%, MBE = −9.33%, RIA = 0.89 and MAE = 27.36 W/m2) while a very good approximation is obtained from the ΚΡ model (R2 = 0.98, RMSE = 19.19%, MBE = −9.80%, RIA = 0.88 and MAE = 31.93 W/m2). In contrast, TD model overestimates λΕM (R2 = 0.96, RMSE = 34.55%, MBE = 15.04%, RIA = 0.78 and MAE = 57.91 W/m2). The results from the analysis of warm season data (also shown in Table 2) indicate that both the PT and the KP model estimates are better, while the TD model estimates are worse.
Fig. 3

Linear regressions between hourly estimated λΕ by TD (λETD) (a) or KP (λEKP) (b) or PT (λEPT) (c) model and measured λΕ (λΕM)

Table 2

Results of the linear regressions [slope (a) and determination coefficient (R2)] and ‘difference measures’ [root mean square error (RMSE), mean bias error (MBE), refined index of agreement (RIA) and mean absolute error (MAE)] between estimated by TD or KP or PT methods and measured hourly actual evapotranspiration, for all or warm period data and for all or warm period data with |β| < 0.3







MAE (W/m2)

All Data






















Warm Period Data






















All Data (|β| < 0.3)






















Warm Period Data (|β| < 0.3)






















It is well established, that as water availability decreases, the energy partitioning between λE and H is altered, resulting in an increase in H, and consequently in β. Thus, β has been sometimes used as an indicator of water stress (Peterschmitt and Perrier 1991; Frangi et al. 1996). The criterion |β| <0.3 is usually used, when applying ΚΡ or TD models, in order to exclude water stress situations (Alves and Pereira 2000). In this study, hourly λETD or λEKP or λEPT estimates based on all- or warm period- data satisfying the above criteria are also calculated and compared with λΕM. The results are shown in Table 2. Hourly λETD estimates from all periods data seem to be slightly improved (R2 = 0.90, RMSE = 32.35%, MBE = 12.67%, RIA = 0.80 and MAE = 53.57 W/m2) indicating that the insufficiency of water rather enhances the overestimation of λΕM by the TD model.

The TD model based on a totally theoretical model for the determination of the rc, compared to the KP model, which uses a semi-empirical approach, employs a different parameterization of ra, as well, in order to estimate λETD. The KP model uses eq. 7 for obtaining raKP, while the TD model uses eq. 8, for calculating raTD. To investigate the effect of the different form of ra, employed by the TD model, λETD was recalculated by employing raKP instead of raTD and the new λETD was compared to λΕM. The results (not shown in Table 2) found from the linear regressions and the ‘difference measures’ for all data (R2 = 0.95, RMSE = 61.29%, MBE = 35.89%, RIA = 0.60 and MAE = 102.56 W/m2) or for the warm period data (R2 = 0.88, RMSE = 65.89%, MBE = 51.35%, RIA = 0.47 and MAE = 151.45 W/m2) show an even greater overestimation of λΕM from this TD approach. Therefore, it is noted that the use of a different form of ra is not responsible for the overestimation of λΕM by the TD model and the cause must be searched in the different parameterization of rc used by the two models.

3.1.3 Effect of Vapor Pressure Deficit and Wind Speed on the Estimation of Canopy Resistance and Actual Evapotranspiration by the KP and TD Models

Canopy resistance is primarily a function of D, U, Rn and water status of a crop (Katerji et al. 2011; Alves and Pereira 2000). Hence, in this part of the study, cases with maximized λΕ (being of great practical interest) and availability of water were considered in order to investigate the effect of D or U on the estimation of rc or λΕ by the KP and TD models. Thus, only hourly data corresponding to high radiation values (Rn > 500 W / m2) and high availability of water (|β| < 0.3) were taken into account. The experimental rcM values were obtained by inversing eq. 3 and substituting as λE, the hourly λΕM values measured by the BREB method.

In order to investigate the effect of U on rc, values of rcM, rcKP and rcTD, corresponding to certain intervals of D {[0–2] or [2–4] or [4–6] kPa} are presented, as a function of U in Fig.4. It is evident that rcM is underestimated from rcTD especially in the larger D range. Similarly, for investigating the effect of D on rc, values of rcM, rcKP and rcTD, corresponding to certain intervals of U {[0–2] or [2–4] or [4–6] m/s} are shown against D in Fig.5. It is apparent that the underestimation of rcM from rcTD increases with the increase of D.
Fig. 4

Canopy resistance (rc) [either measured (rcM) or estimated by the TD (rcTD) or KP (rcKP) model], as a function of wind speed (U) at the reference height z = 6 m, for different ranges of vapor pressure deficit (D). (All presented parameters are based on data with Rn > 500 W/m2 and Bowen ratio| β | <0.3)

Fig. 5

Canopy resistance (rc) [either measured (rcM) or estimated by the TD (rcTD) or KP (rcKP) model], as a function of vapor pressure deficit (D), for different ranges of wind speed (U) at the reference height z = 6 m. (All presented parameters are based on data with Rn > 500 W/m2 and Bowen ratio| β | <0.3)

Furthermore, the estimated values of λETD and λEKP were compared with λΕM and the resulted MBEs for various ranges of D as a function of U or for various ranges of U as a function of D are presented in Fig. 6 (a and b, respectively). It is apparent (Fig. 6a) that the overestimation of λΕM from the TD model increases with increasing U, being rather stabilized for large U values. The underestimation of λΕM from the KP model is small and it is reduced when U becomes large. When the effect of increasing D is considered (Fig. 6b), the overestimation of λΕM from λETD seems to be increased, while the underestimation of λΕM from λEKP is almost diminished.
Fig. 6

Mean bias error (MBE) obtained from the comparisons between the estimated λΕ by TD (λETD) or KP (λEKP) model and the measured λΕ (MBE λETD or MBE λEKP, respectively), as a function of wind speed (U) for certain ranges of vapor pressure deficit (D) (a) or as a function of D for certain ranges of U (b). (The comparisons are applied on data with Rn > 500 W/m2 and Bowen ratio| β | <0.3)

3.1.4 Effect of Vapor Pressure Deficit and Soil Moisture on the Priestley-Taylor Parameter

In order to investigate any dependence of the Priestley-Taylor parameter α upon D or soil moisture, hourly experimental values of α were obtained from the reversal of the eq. 10, when substituting as λE, the hourly λΕM values. The analysis was applied on all hourly data measured during the year 2011, which was selected as a year with no missing data.

The values of parameter α, as a function of D or the volumetric soil moisture (SM) at 10 cm depth are presented in Fig. 7 (a and b, respectively). It is evident that the PT parameter tends to decrease sharply when D increases up to 2 kPa and rather slightly for values of D greater than 2 kPa. In contrast, parameter α does not seem to be affected by SM. It must be noticed that the same result is obtained, when taking into account SM at 20, 30, 40, 60, 80 and 100 cm depth.
Fig. 7

Priestley-Taylor parameter (α) as a function of vapor pressure deficit (D) (a) and volumetric soil moisture content (SM) measured at 10 cm depth (b)

3.2 Diurnal and Annual Variation of Canopy Resistance and Actual Evapotranspiration

Average hourly values of rcM, rcTD and rcKP or λEM, λEΤD, λEΚΡ, λEΡT and Rn were calculated from data corresponding to twenty clear days, [selected as ensuring maximum λΕ values (1–12 July 2011)] and their diurnal variation, presented in Fig. 8, was investigated. Figure 8a illustrates that the diurnal variation of rcKP or rcTD exhibits a similar pattern during the day, with the experimental rcM pattern. All rc values decrease from sunrise, approaching a minimum value at around 8:00 to 10:00 h and then rise steeply from around 15:00 h. The values of rcTD are always smaller than the corresponding values of rcM and rcKP, during the day. The estimated values of rcKP approximate rcM values very well. It is apparent (Fig. 8b) that the diurnal course of λΕM or λEΤD or λEΚΡ or λEΡT shows a bell-shaped curve, similar to that of Rn. λΕ increases sharply after sunrise and takes its maximum value at noon, decreasing later smoothly and then more steeply until the afternoon. Measured or estimated λΕ show an increase at around 13:30 h.
Fig. 8

Diurnal variation of hourly measured (rcM) or estimated by TD (rcTD) or KP (rcKP) model canopy resistance (a), and hourly measured (λΕM) or estimated by TD (λETD) or KP (λEKP) or PT (λEPT) model λΕ and net radiation (Rn) (b). [All parameters are presented as average hourly values, obtained from data of twelve clear days (1–12 July 2011)]

Ten days moving average values of rcM, rcTD and rcKP or λEM, λEΤD, λEΚΡ, λEΡT and Rn at 12.00 am were calculated and their annual variation illustrated in Fig. 9 was examined. According to the annual course of rcM or rcTD or rcKP (presented in Fig. 9a), the underestimation of rcM from rcTD is large during the warm period, while a high agreement between rcKP, and rcM is evident. During the cold period, rcKP seems to slightly overestimate rcM. Figure 9b illustrates that the annual course of λΕM or λEΚΡ or λEΡT traces Rn very well, while λETD deviates from Rn, especially during the warm period. A large overestimation of λΕM by λETD is apparent during the warm period, while a high agreement between λEΚΡ or λEPT and λΕM is evident. During the cold period, λEΚΡ and λEPT slightly underestimate λΕM.
Fig. 9

Annual variation of hourly measured (rcM) or estimated by TD (rcTD) or KP (rcKP) model canopy resistance (a), and hourly measured (λΕM) or estimated by TD (λETD) or KP (λEKP) or PT (λEPT) model λΕ and net radiation (Rn) (b), at 12.00 am. (All parameters are presented as ten days moving averages, obtained from all data of the year 2011)

4 Discussion

4.1 Estimation of Hourly Canopy Resistance and Actual Evapotranspiration

The evaluation of the three methods employed for estimating λΕ over olive trees on hourly basis (Table 2), suggests that PT and KP models are the most appropriate (RIA equal to 0.89 or 0.88, respectively) followed by the TD model (RIA = 0.78). These results are in agreement with those reported by Shi et al. (2008), when the three methods were used over a forest ecosystem, on a daily scale. PT or KP model underestimates λΕ by 9.3% or 9.8%, respectively, while TD model overestimates λΕ by 15.0%, increased up to 25.8%, during warm period.

This study shows that the KP model seems to be a robust and reliable model for predicting λΕ and this has already been reported for other cultivated or natural crops, subjected to contrasting water conditions (Katerji and Rana 2006; Katerji et al. 2011; Lecina et al. 2003; Pauwels and Samson 2006; Rana et al. 1994; Rana et al. 2005; Rana and Katerji 2008; Shi et al. 2008; Steduto et al. 2003; Zhu et al. 2014; Rana et al. 2012; Li et al. 2015; Ayyoub et al. 2017; Amazirh et al. 2017).

On the other hand, the use of TD model has been shown as not recommended for the estimation of λΕ above olive trees. The model has already been referred as not suitable over other crops (overestimating up to 30–50%). It has been though referred as underestimating λΕ above irrigated grass, by about 20% (Katerji et al. 2011). During conditions of no water stress and maximized λΕ, it may be used only for low values of D or low values of U [Fig. 6 (a, b)]. The disagreement between estimated by the TD model and measured λΕ is mainly due to the parameterization of rc by the model. rcTD is mainly a function of D, but the variables associated with soil and plant water deficit are not taken into account and consequently, its value is assumed to be the same for all surfaces, which is not applicable (Katerji et al. 2011). In this study, this is evident from the increasing MBE of the model, as D increases (Fig. 6b), and the especially great disagreement between λETD and λΕM, during the warm period, which is usually associated with higher values of D (Table 2). It is also supported from a better agreement between λETD and λΕM, when the data from no water stress cases are considered in the analysis (Table 2). When D increases, higher values of measured rc are expected (due to stomata closure), as already reported over olive trees (Bongi and Loreto 1989; Fernandez et al. 1993, 1997) or other crops (El-Sharkawy and Cock 1984; Hernández et al. 1989; Mansfield and Davies 1981; Schulze et al. 1972; Schulze and Hall 1982). Great increases of wind velocity have been also referred, as furthermore speeding up the stomata closure caused by D increases, through a feedback mechanism (El-Sharkawy 1990) and thus resulting in increasing measured rc. The effect of ra on λΕ through rc has also been discussed (Alves and Pereira 2000; Katerji et al. 2011), but the parameterization of rcTD does not incorporate the effect of ra. This results in greater underestimations of experimental rcM (and greater overestimations of λE, respectively) by the TD model, when U increases and thus ra decreases. This is shown in Fig. 6a by the increasing MBE of the λETD predictions, when U increases. It must be also noticed that the employment of a different ra (as the one used by the KP model) in the analysis, resulted in worse predictions (RIA = 0.60) and an even greater overestimation (MBE = 35.9%) in agreement with the findings presented by Katerji et al. (2011) for other crops.

In this study, the PT model has been evaluated as a reliable model for predicting λΕ over olive trees, when based on the calibration of its parameter α during the warm and cold period. The parameter has been referred as depending upon D (Shi et al. 2008) and SM (Flint and Childs 1991; Fisher et al. 2005), but in the present study the experimental values of α were found as reducing when D increases but did not appear to be dependent on SM [Fig. 7 (a, b)]. These results are quite in agreement with findings presenting a greater dependence of α on D and a small dependence on SM, as referred for a forest by Shi et al. (2008). They are though, in coincidence with part of the results reported above a natural meadow by Pauwels and Samson (2006), who found that the parameter α decreases with the increase of D but increases with the increase of the soil moisture. Probably ‘the weak influence of soil moisture on α’ found in this work may be due to the absence of significant water stress during the study period as it has also been noticed by Shi et al. (2008).

4.2 Diurnal and Annual Variation of Canopy Resistance and Actual Evapotranspiration

The diurnal variations of predicted and measured λΕ follow the bell-curved shape, similarly to net radiation (Fig. 8b). A rapid increase in predicted and measured λΕ has been observed shortly after sunrise, reaching its peak at noon of solar time and followed by a slow decline at first and a more rapid one over the afternoon. However, λΕM is significantly overestimated by the TD model at noon, when high values of D are expected. Similar patterns of daily variation of λΕM were also reported over olive orchard (Villalobos et al. 2000; Berni et al. 2009), over alpine grass (Zhu et al. 2014) and over forest ecosystem (Shi et al. 2008).

The diurnal course of rcM, rcTD and rcKP over olive trees (Fig. 8a) shows that the experimental rcM and the estimated rcTD and rcKP attain similar variation, with the rcKP values being in a very good agreement with rcM values. However, rcTD values are always significantly lower than those of rcM and rcKP, indicating that the TD model is rather unreliable over olive trees. The observed in this analysis, diurnal evolution of rc is similar to the ones referred by other studies over olive orchard crops (Alves and Pereira 2000; Katerji and Rana 2006) and the ones reported over forest (Shi et al. 2008), irrigated grass (Rana et al., 1994) and alfalfa (Lascano et al. 2010).

According to the annual variation of rcM, rcKP and rcTD, the underestimation from the TD model is large, during the warm period, while there is a high correlation between rcKP and rcM. The annual course of estimated λEΚΡ, λEΡΤ and λΕM traces the Rn. These results are in agreement with the findings reported by Er-raki et al. (2008) or Martinez-Cob and Faci (2010) over olive trees, when using λΕ measured by Eddy covariance or estimated by PM method. As far as λEΤD concerns, its annual course seems to be particularly affected by the annual course of D. During the warm period, the overestimation of λΕM by TD model is large, while there is a high coincidence between λE and λΕM. During the cold period the ΚP model underestimates λΕM.

5 Conclusions

The use of the Penman-Monteith equation for estimating λΕ is mainly limited by the difficulty of determining rc. In this study, rc was parameterized by a semi-empirical approach (KP model) or a theoretical approach (TD model) or was considered as zero (PT model), in order mainly to evaluate the applicability of the three models in estimating accurate hourly λΕ over olive trees. The conclusions are as follows:
  • The PT model (when based on the calibration of its parameter α during the warm and cold period) and the KP model are the most appropriate (RIA equal to 0.89 or 0.88, respectively), followed by the TD model (RIA = 0.78). PT or KP model underestimates λΕM by 9.3% or 9.8%, respectively, while TD model overestimates λΕM by 15.0%, increased up to 25.8%, during warm period.

  • In conditions of maximizing λΕ, the KP model satisfactorily evaluates λΕM while the TD model overestimates it. The overestimation of TD increases with the increase of D, with the increase of U and when the sufficiency of water is not ensured.

  • The experimental rcM and the estimated rcTD and rcKP attain similar diurnal variation. The rcKP values are in a very good agreement with rcM values, while rcTD values are always significantly lower than those of rcM and rcKP, indicating that the TD model is rather unreliable over olive trees. This is also evident from the annual variation of rcM, rcKP and rcTD. The underestimation from the TD model is large, during the warm period, while there is a high coincidence between rcKP and rcM.

  • The experimental values of the PT parameter α were reducing with D increasing but they were not found as influenced by SM.



The National and Kapodistrian University of Athens, the Greek Ministry of Rural Development and Food and the former Prefecture of Laconia for funding the project are duly acknowledged. A previous shorter version of the paper has been presented in the 10th World Congress of EWRA “Panta Rei” Athens, Greece, 5-9 July 2017 and has been published in the European Water Journal (Margonis et al. 2017)

Compliance with ethical standards

Conflict of Interest



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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Athanasios Margonis
    • 1
  • Georgia Papaioannou
    • 1
    Email author
  • Petros Kerkides
    • 2
  • Gianna Kitsara
    • 1
  • George Bourazanis
    • 2
  1. 1.National and Kapodistrian University of Athens, Department of PhysicsAthensGreece
  2. 2.Agricultural University of Athens, Department of Natural Resources Development and Agricultural EngineeringAthensGreece

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