Introduction

Evapotranspiration (ET) is an important parameter needed for the investigation of water and energy balances over naturally occurring vegetative surfaces, with applications in agrometeorology and climatology (Martano 2015). Accurate estimation of ET has an important significance to study the ecosystem productivity and water balance in order to guide irrigation, agricultural drainage and to improve agricultural water use. In general, several approaches have been undertaken to define the concept of ET which at best distinguished between potential, actual and reference ET (Thornthwaite 1948; Allen et al. 1998; Hansen et al. 1980; Dingman 1994; Watson and Burnett 1995; Jensen et al. 1990). Direct techniques for measuring ET includes lysimetry (Howell et al. 1991; Zhang et al. 2016), pan evaporimeter (Irmak et al. 2002), scintillometry (Daoo et al. 2004) and eddy covariance (Baldocchi et al. 2001). However, the direct methods have been found to be highly capital intensive, laborious to setup and requires intensive maintenance (Watson and Burnett 1995; Odi-Lara et al. 2016) in addition to certain drawbacks in their operation mechanism. For instance, the lysimetry technique does not maintain original soil profile characteristics including density during and after construction (Allen et al. 2011). Evaporation pans measure the loss of a known quantity of water through evaporation using a vernier caliper to identify changes in the precise water level, but they do not measure transpiration. To compensate, pan evaporation data must be adjusted downward using a pan coefficient. Pan evaporation data typically overestimates ET due to the nature of the pan. Evaporation pans often will still evaporate at night due to the residual heat that is stored in the water of the pan. Hence, researchers have developed empirical models which require measurements of routine meteorological parameters as in-data to estimate ET over the years (Martano 2015). These routinely measured parameters tend to be relatively easy to collect, they include; temperature, relative humidity, rainfall, wind, and solar radiation (Maselli et al. 2014). ET models have been categorized into three basic types: temperature, radiation, and combination (Jensen et al. 1990; Watson and Burnett 1995). Temperature models generally require only measurements of air temperature as the sole meteorological input to the model (e.g., Thornthwaite 1948; Doorenbos and Pruitt 1977; Jensen et al. 1990). Radiation models (e.g., Doorenbos and Pruitt 1977; Hargreaves and Samani 1985) are typically designed to use some component of the energy budget concept and usually require some form of radiation measurement. Finally, combination models combine elements from both the energy budget and mass transfer models to give very accurate results (Jensen et al. 1990). The Penman family of models is the most common combination model in use today (Jensen et al. 1990; Allen et al. 1998).

This study thus compares direct measurement of ET from eddy covariance technique to ET estimates obtained from four standardized models namely; FAO-56 Penman–Monteith, Priestly-Taylor, Makkink and Turc at Ile-Ife, a tropical location. These are with a view to evaluating the performance of the empirical models and validate their suitability for agro-meteorological purposes in Nigeria.

Materials and methods

Site description

The experimental site for this study was located at the Sports Complex (7°33′N, 4°35′E), Obafemi Awolowo university campus, Ile-Ife, southwestern Nigeria (Fig. 1). The climate of the study area is generally characterized with two seasons: namely the wet and dry season, the wet season spans between March to October and the dry season lasts from November to February. The variation of this season is a function of the meridional movement of the Inter-Tropical Discontinuity (ITD) which demarcates the warm and moist South-Westerly trade wind at the surface from the hot and dry North-Easterly trade winds (Jegede et al. 2006). The onset months of the wet season falls within the weather zone B (extends 200–400 km south of the surface position of the ITD).The zone is characterized by suppressed convection resulting into cumulus clouds and precipitation is limited to light showers, whereas in August/September, which is at the peak of wet season in the area, Ile-Ife falls within the weather zone D characterized by stratus cloud and is accompanied by light rains and drizzles with occasional moderate thunderstorm activities (Ayoola et al. 2014). The field measurements for this study took place between 31st of May and 14th of June, 2013.

Fig. 1
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Map showing the evapotranspiration measurement site within Obafemi Awolowo University, Ile-Ife (insert is the map of Nigeria indicating the location of Ile-Ife)

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Data acquisition

A measurement complex consisting of two but closely located masts (2.5 and 2.8 m) were setup over a plane grass-covered surface. An eddy covariance (EC) system comprising of a CSAT3 sonic anemometer and a LI-7500 (fast response) open path infrared gas analyzer was installed on the 2.5 m mast to measure the vertical transport of water vapour flux over the grass covered surface (Fig. 1). The 2.8 m mast was instrumented with an array of slow-response and well calibrated sensors (NRLITE, CS300, HMP45, ML101) deployed for routine observation of meteorological parameters such as net radiation, global radiation, air temperature, relative humidity, wind speed (Fig. 2). The EC system measures the latent heat (LE) and water exchange between the ground surface and the atmosphere. The rate of evapotranspiration is then determined by converting measured latent heat flux from the EC as follows

Fig. 2
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Setup of a the Eddy Covariance system comprising of a CSAT 3 sonic anemometer and a Li-7500 infrared gas analyzer b slow response measurement complex for microclimatic data acquisition at the experimental site in Ile-Ife

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$$ET~=\frac{{LE}}{{{\rho _w}\; \times \;\lambda }},$$
(1)

where \(LE\) is the Latent heat flux (W/m2), \({\rho _w}\) is the density of water (Kgm−3) and \(\lambda\) is the latent heat of vaporization (MJkg−1).

The ET models employed

  1. i.

    The FAO-56 Penman–Monteith employed in this study is given

$$ET=\frac{{0.408\Delta \left( {{R_n} - G} \right)+\frac{{\gamma 600{U_2}\left( {{e_s} - {e_a}} \right)}}{{{T_a}+273}}}}{{\Delta +\gamma \left( {1+0.56{U_2}} \right)}},$$
(2)

where \({R_N}\) is net radiation, G is ground heat flux, U2 is wind speed at 2 m/s, \({e_s} - {e_a}\) is vapour pressure deficit, \({T_a}\) is air temperature, γ is the psychometric constant and Δ is the slope of saturation vapour pressure curve. The FAO-56 PM was adjusted so as to locally account for the influence of the bulk and aerodynamic resistances which are weather dependent parameters and as a result they vary for different climatic conditions (See Allen et al. 1998).

  1. ii.

    The Priestly Taylor Model

The Priestly and Taylor (1972) model used in this study is given as

$$ET= \propto \frac{\Delta }{{\Delta \;+\;\gamma }}({R_n} - G),$$

\(\propto\) = 1.26, and all other terms are as defined previously.

  1. iii.

    The Makkink Model:

The Makkink model (1957) developed in the Netherlands was also used, and the model is given by

$$ET=0.61\frac{\Delta }{{\Delta +\;\gamma }}\frac{{{R_s}}}{{2.45}} - 0.12,$$
(4)

where \({R_s}\) is the incoming solar radiation, γ is the psychometric constant and Δ is the slope of saturation vapour pressure curve.

  1. iv.

    The Turc Model:

The Turc model (Turc 1961) has two forms depending on the consistent value of the relative humidity. Since RH > 50% for this study area, the form of the model used in this work is

$$ET=0.013{T_a}\; \times \;\left( {23.9{R_s}+50} \right)/\left( {{T_a}+15} \right).$$
(5)

All the terms and variables are as earlier mentioned.

Models’ performance evaluation

Comparative evaluation of model performance was done using both statistical criteria (quantitative) and graphical display (qualitative) (Loague and Green 1991). The combined approach is useful in making comparative evaluations of model performance between alternative or competing models. In this work; the correlation measure R, the Mean biased error (MBE), the Root mean square error (RMSE) and the index of agreement (d) were calculated and reported. It was pointed out (Fox 1981; Willmott 1982) that the commonly used correlation measure such as \(R~\) and \({R^2}\) are often inadequate when used to compare model predicted and observed variables. This is because the magnitude of \(R\) is not related to the accuracy of model prediction (Willmott et al. 1985). As a result of this, recommendations have been made to use other sum and difference measures to confirm the accuracy of the predictions (which consequently influence the decision to use MBE, RMSE and d in this study Krause et al. 2005; Licciardello et al. 2007; Carrillo-Rojas et al. 2016; Odi-Lara et al. 2016). Normally, the coefficient \(R\) has values ranging from 0 to 1; the closer to 1 the stronger the correlation and vice versa, a value of zero indicates no correlation between the observed and predicted values. Moreover, the MBE describe biases between the observed and the predicted values, the RMSE expresses the average difference between observed and predicted values both have values ranging from −\(\infty\) to\(\infty\). Positive values of these statistical measures indicate overestimation while their negative values signify underestimation. The lower the error estimate the better the prediction by the models.

Models’ recalibration

The empirical models employed to estimate ET were developed in regions with differing climatic regimes. Tabari and Talaee (2011) reported that numerous scientists working on ET under different climates have shown that the models vary in their performance spatially and temporally. The implication of this is that these models require local calibration for effective ET estimation when adapted to other locations (Allen et al. 1998; Pereira et al. 1999). ET models are therefore often calibrated against a more reliable method of estimation with a view to improving their accuracy in estimating ET (Fontenot 2004). In this study, the adopted models namely the PM, the PT, the MK and the TURC models were calibrated on the basis of the EC method. The main approach for calibration of the models used in this study is linear regression with EC measurement. According to Houshang et al. (2012), the intercept, a, and slope, b, of the best fit regression line is then used as regional calibration coefficients:

$$E{T_{cal}}=a+b\left( {E{T_{model}}} \right).$$
(6)

Results and discussion

The results of the mean daytime evapotranspiration measured by the eddy covariance technique and the estimation of reference evapotranspiration models over the research site for this study are presented in the Table 1. 12 fair-weather days were presented for discussion as other days were considered spurious due to precipitation and condensations which were the prevalent weather conditions of those days. From the Table 1, the daily EC measurement range between 0.45 and 1.05 mm/day, the ET as estimated by PM range between 0.49 and 1.27 mm/day, the PT model give a range of value between 0.91 and 2.22 mm/day, while the MK model range between 0.74 and 1.76 mm/day and the TURC model give a range of value between 0.79 and 1.88 mm/day. Figure 3(a–l) showed the diurnal variation of measures and estimated ET as given by each of the model. Significant evapotranspiration does not occur in the nocturnal periods due to the fact that there is no sufficient energy from the incoming solar radiation and wind to drive the process. An observation of the range of values of ET estimate presented in Table 1 gave a clear indication that the PM model provided the closest values to those of the EC measurement chosen as the reference method while the PT model gave the farthest estimates. It is also obvious that the MK model and the TURC model estimated closer to the EC in that order. The excellent performance of the PM model can be attributed to the fact that it requires many meteorological data such as radiation, wind speed, air temperature and humidity (Tabari and Talaee 2011). These weather parameters were so fundamental to the occurrence of evapotranspiration process. The observed overestimating performance of the PT model has been well reported in literatures (Panda and Kashyap 2001; Fontenot 2004). In order to investigate the relationship between the ET models and the directly measured evapotranspiration data, a regression analysis was carried out and the standard error of estimate (SEE) was obtained. The result is presented in Table 2. All the models show strong correlation with the EC measurement on the account of the coefficient of determination (R2 > 0.9) which describes colinearity between observed (O) and predicted (P) variates where the observations are interval or ratio in scale (Willmott 1981). However, Variance in their SEE and slope can be used to evaluate their respective performances. The result in Table 2 show that the PM model has the lowest error of estimate of 0.049 and a slope closest to unity of 0.687 hence it has the strongest performance correlation with the benchmark measurements. The MK model followed the PM model in performance with the second lowest SEE of 0.395 and a slope of 0.569. The TURC model rated 3rd in the correlation performance with SEE of 0.553 and a slope of 0.539. The PT model has the highest error of estimate (1.32) and the lowest slope (0.386). Table 3 showed the percentage departure of ET values obtained by the model from the EC measurement. From the table the FAO-56 Penman–Monteith (PM) model has the least departure from the benchmark measurement with an average overestimation of 14.7%. The Makkink (MK), Turc and Priestly-Taylor (PT) models overestimated the EC measurement by 51.1, 60.7 and 94.1% respectively. Figures 4 and 5 were presented to examine the influence of surface radiative fluxes on the performances of the ET methods. As can be seen from the time series, the fluxes were the major determinant of the variation and rate of ET occurrence. The peaks, dips and the fluctuations in the fluxes over a diurnal course were quite identical. The Julian days 159, 160 and 162 were of particular interest as dips in net and global radiation fluxes occasioned low estimates of evapotranspiration rates. In order to improve the performances of the models, 10 days estimate were used to obtain recalibrated coefficients. The result of the ET estimate for Julian days 163 and 164 were then used to validate the procedure and examine the performance of the adjustment. The result is presented in Fig. 6. The results of coefficient adjustment presented graphically above show an improvement in the performances of these models. It can be seen that the PM model does improve after recalibration and similarly the other models. The adjustment almost brought the MK and TURC model to the exact estimates of the EC measurement, this implies recalibration has clearly improved the performance of both models and makes them suitable for this study location. In order to ensure a rigorous comparison of the performances of these models before and after recalibration, an extended analysis was performed using three different statistical indices for the estimated values. The result of these statistical tests is presented in Table 4a, b. Intercepts (a) and slopes (b) for the least squared regression analysis also were obtained and reported (Fig. 7). while the adjusted versions of the ET models are presented in Table 5. These results further strengthen the investigation to see the influence of calibration on the performances of these models. As shown above all the statistical measures employed for the models performance evaluation are in agreement with the results obtained by the regression analysis. The errors reduced after adjustment for every one of the models, the index of agreement came closer to unity though there was no much effect on intercept yet the regression analysis show an improved result in ET estimations after the models were calibrated, changing the slopes closer to unity.

Table 1 Daily values of ET from direct measurement (EC) and Models
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Fig. 3
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a–l Diurnal variation of measured and estimated ET over grass at Ile-Ife

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Table 2 Coefficient of linear regression and error estimate for the performances evaluation of the models
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Table 3 Percentage departure of ET estimate by models from ET obtained from Eddy covariance measurement
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Fig. 4
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Period of observable strong dip in net and global radiation fluxes

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Fig. 5
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Models response to dip in radiation fluxes

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Fig. 6
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ET estimates from models before (a) and after (b) recalibration of coefficients

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Table 4 a Statistical analysis of ET models before coefficient adjustment. b Statistical analysis of ET models after coefficient adjustment
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Fig. 7
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Least-square regression analysis of EC Benchmark values and ET model estimates

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Table 5 ET models and their corresponding adjusted versions for the study location
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Conclusion

The conclusions from this study were that the FAO-56 Penman–Monteith equation is suitable for Ile-Ife, a tropical climate with or without adjustment especially if the bulk and aerodynamic resistances were calculated from local weather data. Also, the performance of the other empirical models improved once they were recalibrated by adjusting their original coefficient based on the measured ET values obtained from the EC technique. The improvement in the performance of the models requiring less input data will make irrigation scheduling less rigorous, cost effective and accessible.