Accurately estimating crop evapotranspiration (ET) is essential for agricultural water management in arid and semiarid croplands. This study developed extreme learning machine (ELM) and generalized regression neural network (GRNN) models for maize ET estimation on the China Loess Plateau. Maize ET, meteorological variables, leaf area index (LAI), and plant height (hc) were continuously measured during maize growing seasons of 2011–2013. The meteorological data and crop data including LAI and hc from 2011 to 2012 were used to train the ELM and GRNN using two different input combinations. The performances of ELM and GRNN were compared with the modified dual crop coefficient (Kc) approach in 2013. Results indicated that ELM1 and GRNN1 using meteorological and crop data as inputs estimated maize ET accurately, with root mean square error (RMSE) of 0.221 mm/d, mean absolute error (MAE) of 0.203 mm/d, and NS of 0.981 for ELM1, RMSE of 0.225 mm/d, MAE of 0.211 mm/d, and NS of 0.981 for GRNN1, respectively, which confirmed better performances than the modified dual Kc model. Performances of ELM2 and GRNN2 using only meteorological data as input were poorer than those of ELM1, GRNN1, and modified dual Kc approach, but its estimation of maize ET was acceptable when only meteorological data were available.

INTRODUCTION

Population growth and increasing consumption of calorie- and meat-intensive diets are expected to roughly double human food demand by 2050 (Mueller et al. 2012). To meet this increasing food demand in the coming decades, new practices for agricultural water management must be developed, especially in arid and semiarid regions, to boost crop production per amount of water use, i.e., crop water use efficiency (WUE). As the only term that appears in both water balance and surface energy balance equations (Xu & Singh 2005), evapotranspiration (ET) is not only the basis of a deep understanding of ecological and hydrological processes, but also an important indicator to evaluate WUE in agriculture. About half the lands on Earth are short of water (Newman et al. 2006), and more than 90% of water used in agriculture is lost by ET (Rana & Katerji 2000; Ding et al. 2013). Therefore, accurate estimation of ET is of importance to regional agricultural water management aiming at water saving and an increase of WUE (Zhang et al. 2013).

In the past decades, numerous methods, which can be grouped into single-layer (e.g., Penman–Monteith), two-layer (e.g., Shuttleworth–Wallace) and multi-layer (e.g., Clummping) models (Monteith 1965; Shuttleworth & Wallace 1985; Brenner & Incoll 1997), have been proposed for ET estimation since direct measurement of ET is difficult, costly, and not available in many regions (Allen et al. 1998; Ding et al. 2013). The main limitation of these methods developed on a physical basis, however, is their required input data cannot be easily measured, such as aerodynamic resistance and surface resistance (Allen 2000; Ding et al. 2013). To overcome this deficiency, the indirect FAO-56 dual crop coefficient approach was proposed by Allen et al. (1998), which was the product of reference evapotranspiration (ET0) and crop coefficient (Kc). ET0 is the evapotranspiration rate of the reference crop with an assumed crop height of 0.12 m, a fixed surface resistance of 70 s/m, and an albedo of 0.23, while Kc, the ratio of ET and ET0, represents the effects of characteristics that distinguish the specific field crops from the reference crop. Compared with the single Kc method, the dual Kc approach makes it possible to better assess the impacts of soil wetting by rain or irrigation, as well as the impacts of keeping part of the soil dry or using mulches for controlling soil evaporation (Zhang et al. 2013), and has been widely applied for ET estimation due to the simplicity and good performances of the approach (Allen 2000; Rousseaux et al. 2009; Liu & Luo 2010; Ferreira et al. 2012; Zhao et al. 2015). Although good performances of this approach have been widely reported, the straightforward adoption of generalized crop coefficients recommended by FAO-56 can lead to errors in the estimation of ET since the dividing of crop growth period and associated crop coefficients are closely related to local climate and crop conditions (Katerji & Rana 2006; Poblete-Echeverría & Ortega-Farias 2013; Zhao et al. 2015). Thus, a modification of the dual Kc approach is needed when applying the method for ET estimation.

In recent years, artificial intelligence (AI) has been successfully implemented in ET0 estimation. The application of AI in ET0 modeling was first investigated after 2000 by Kumar et al. (2002), who estimated ET0 using an artificial neural network (ANN). Since then, efforts towards the estimation of ET0 using ANN models have been performed (Trajkovic et al. 2003; Kisi 2006; Kim & Kim 2008; Landeras et al. 2008; Traore et al. 2010; Martí et al. 2011a). More recently, many studies have proposed some new AI approaches for ET0 estimation: Tabari et al. (2012) investigated the performances of support vector machines, adaptive neuro-fuzzy inference system for ET0 estimation in a semi-arid highland environment in Iran; Shiri et al. (2011, 2012) applied a genetic programming approach for ET0 and evaporation modeling; Kisi et al. (2012) developed generalized neurofuzzy-based evaporation models in Arizona, USA; Pour Ali Baba et al. (2013) estimated ET0 using adaptive neuro-fuzzy inference system and ANN for two weather stations in South Korea; Kisi (2013) investigated the applicability of Mamdani and Sugeno fuzzy genetic approaches in modeling ET0 in Turkey; Shiri et al. (2013) and Martí et al. (2015) used gene expression programming for ET0 estimation; Kisi (2016) investigated the ability of least square support vector regression, multivariate adaptive regression splines and M5 Model Tree in modeling ET0 in Turkey; Abdullah et al. (2015) and Feng et al. (2016) examined the capability of extreme learning machine (ELM) for ET0 estimation in Iraq and Southwest China, respectively. All these studies confirmed good performances of AI approaches for ET0 estimation worldwide. Differently to ET0, which is from a reference crop and only affected by meteorological variables, ET for a specific crop is more complicated and affected not only by meteorological variables but also soil properties, crop characteristics, and agronomy management. To the best knowledge of the authors, there are no former studies evaluating the performances of AI approaches for rainfed maize ET estimation considering experimental data as targets.

This study applied two AI approaches, i.e., ELM and generalized regression neural network (GRNN), for maize ET estimation using meteorological and crop data on the China Loess Plateau. The performances of ELM and GRNN were assessed against the modified dual Kc approach for evaluating the newly proposed models considering measured maize ET data from eddy covariance systems in a rainfed cropland as benchmark.

MATERIALS AND METHODS

Study site

The study was carried out in a rainfed spring maize field at the Experimental Station of Dryland Agriculture and Environment (ESDAE), Ministry of Agriculture, P. R. China, which is located in Shouyang, Shanxi Province, northern P. R. China (37°45′58″N,113°12′9″E, 1,202 m Alt.) during three maize growing seasons (from May 1 to September 28, 2011, May 3 to September 22, 2012, and April 28 to September 25, 2013). The experimental station has a typical continental temperate climate with a mean daily temperature of 7.4 °C, mean annual precipitation of 481 mm and mean annual frost-free days of 140 days. Mean annual precipitation during the growing season of spring maize is about 330 mm. The soil at the experimental station is classified as a cinnamon soil with light clay loam texture and an average bulk density of 1.34 g/cm3. Average volumetric soil water content at field capacity and wilting point were 36.0% and 12.0% to a depth of 1.0 m, respectively. Groundwater is about 150 m below ground surface (Gong et al. 2015).

The maize crop, variety Jingdan-951, was sown in north–south rows with the distance between rows equal to 50 cm and the space between two plants within rows of 30 cm. The maize sowing rate was 66,667 seeds per ha for 3 years. The area of the experiment plots is about 3.0 ha, of which length and width is 200 m and 150 m, respectively, which meets the minimum fetch requirement of eddy covariance system installation (Gong et al. 2015).

Measurements

Meteorological data

Half-hourly meteorological variables were obtained by an automatic weather station (Campbell Scientific Inc., Logan, UT, USA) nearby the experimental plots (Gong et al. 2015). Solar radiation (Rs) was measured with a Silicon Pyranometer (LI200X, LI-COR, Inc., Lincoln, NE, USA) and precipitation (P) was registered with a pluviometer (RGB1, Campbell Scientific Inc.). Wind speed (u2) and its direction (w) were measured using a cup anemometer and a wind vane (03002-L, R. M. Young Inc., Traverse, MI, USA), respectively. Air temperature (T) and relative humidity (RH) were measured using a Vaisala probe (HMP45C, Vaisala Inc., Tucson, AZ, USA). All variables were monitored at 2 m above the surface and recorded in a data-logger (CR10RX, Campbell Scientific Inc.).

Eddy covariance system and evapotranspiration measurements

In the center of the experimental plot, latent (LE) and sensible heat (H) fluxes were measured by an open-path eddy covariance system mounted on a tower, which consisted of an open-path infrared gas analyzer (LI-COR Inc., model LI-7500) and a three-dimensional supersonic anemometer (Campbell Scientific Inc., model CSAT3). A temperature and humidity sensor (Campbell Scientific Inc., model HMP45C), a four-way net radiometer (Kipp & Zonen Inc., Delftechpark, The Netherlands, model CNR1), and self-calibrating heat flux sensors (Campbell Scientific Inc., model HFP01) were also used. The sensor height was adjusted to keep the relative height of 0.5 m between sensors and maize canopy constant at interval of 5–7 days. Specific time length depended on the increments of canopy height. The observation site had a wide fetch of at least 50 m in all directions, which allowed us to neglect heat advection in the maize field (Gong et al. 2015). All of the measured meteorological and fluxes data were averaged at daily timescale in the present study.

Leaf area index and plant height measurements

Five maize plants were randomly selected to measure plants height (hc), leaf length, and width at internal of 1 or 2 weeks during the maize growing season. Specific time length depended on maize growth stages and leaf growth rates. Leaf area index (LAI) was calculated by summing products of lamina length and maximum width of each leaf and then multiplying by an empirical coefficient of 0.74, and then dividing by area per plant (30 cm × 50 cm) (Li et al. 2008; Gong et al. 2015): 
formula
1
where LAI is leaf area index (m2/m2); Li and Wi is the length and width of the i-th leaf, respectively; Drow and Splant stand for the distance between the two rows and the space between the plants in the row, respectively.

The modified dual Kc approach for ET estimation

Reference evapotranspiration

Daily reference evapotranspiration (ET0) was estimated using the FAO-56 Penman–Monteith equation (Allen et al. 1998; Feng et al. 2014): 
formula
2
where ET0 is reference evapotranspiration (mm/d); Rn is net radiation (MJ/m2 d); G is soil heat flux density (MJ/m2 d); Tmean is mean air temperature (°C); es is saturation vapor pressure, kPa; ea is actual vapor pressure, kPa; Δ is slope of the saturation vapor pressure function (kPa/°C); γ is psychometric constant (kPa/°C); and u2 is wind speed at 2 m height (m/s).

Maize ET

Actual maize ET was calculated by multiplying ET0 and Kc (Allen et al. 1998): 
formula
3
According to the dual Kc approach, Kc can be split into two parameters, which are Ke and Kcb. Thus, Equation (3) can be rewritten as: 
formula
4
where ET is crop evapotranspiration (mm/d); Kcb is basal crop coefficient; Ke is soil evaporation coefficient; Ks is water stress coefficient. Thus, KsKcbET0 represents crop transpiration while KeET0 represents soil evaporation.
  • (1) Calculation of Kcb:

According to FAO-56, the original Kcb is calculated as (Allen et al. 1998): 
formula
5
where Kcb,o is the original basal crop coefficient obtained; Kcb(Tab) is basal crop coefficient value taken from FAO-56 recommendation; u2 is wind speed at 2 m height (m/s); RHmin is minimum daily relative humidity (%); and hc is crop height (m).
This study applied canopy cover coefficient (Kcc) to accurately calculate daily actual Kcb (Ding et al. 2013): 
formula
6
where Kc,min is the minimum basal crop coefficient for bare soil; Kcb,full is basal crop coefficient when crop has nearly full ground cover. Kcb,full can be calculated following the method of Allen et al. (1998): 
formula
7
Kcc was then calculated by the ratio of radiation intercepted by crop canopy: 
formula
8
where κ is the coefficient for radiation extinguish and LAI is leaf area index.
  • (2) Calculation of Ke:

Ke is calculated by estimating energy availability and soil moisture regimes at the soil surface through the original FAO-56 procedure: 
formula
9
where Ke is the soil evaporation coefficient; Kc,max is the maximum value of Kc following rain or irrigation; Kr is evaporation reduction coefficient; few is the fraction of the soil that is both exposed to solar radiation and wetted (0.01–1.0). Evaporation is restricted by the energy available at the exposed soil fraction, i.e., Ke could not exceed few Kc,max.
Kc,max is adjusted for crop height and climate: 
formula
10
 
formula
11
 
formula
12
where fc is the approximate fraction of soil surface that is exposed; fw is the average fraction of soil surface wetted by irrigation or precipitation; De,i−1 is cumulative depth of evaporation from the soil surface layer at the end of day i−1 (mm); TEW is total evaporable water (mm); REW is readily evaporable water (mm). When De,i-1REW, Kr = 1.0. 
formula
13
where θFC and θWP are soil water contents at field capacity and wilting point, respectively (m3/m3). Ze is depth of the topsoil layer.
fc was determined by LAI: 
formula
14
De calculation requires a daily water budget for the exposed and wetted surface. 
formula
15
where De,i and De,i-1 are cumulative depth of evaporation of the topsoil at the end of day i and i−1 (mm); Pi is precipitation on day i (mm); ROi is runoff from the soil surface on day i (mm); Ei is evaporation on day i (mm); Tew,i is depth of transpiration of the soil surface layer on day i (mm), Tew,i = 0 for row crops as recommended in Allen et al. (1998); DPe,i is deep percolation loss from the topsoil layer on day i if soil water content exceeds field capacity (mm).
  • (3) Calculation of water stress coefficient: 
    formula
    16
where Dr,i−1 is root zone depletion at the end of day i−1 (mm); TAW is total available water in the root zone (mm); RAW is the readily available soil water in the root zone (mm). When Dr,i-1 < RAW, Ks = 1.0. 
formula
17
where Zr is the rooting depth (m).

AI models for ET estimation

Extreme learning machine

The ELM was first proposed by Huang et al. (2006). Its learning speed can be thousands of times faster than traditional feedforward neural network (FFNN) learning algorithms such as back-propagation algorithm while obtaining better generalization performance. For traditional FFNN, all the parameters need to be tuned and thus there exists a dependency between different layers of parameters (weights and biases) (Huang et al. 2006). The advantage of ELM is that the hidden layer does not need to be tuned and the learning speed is faster than traditional FFNN, in addition to better generalization performance (Abdullah et al. 2015). The ELM has successfully been applied for function approximation in hydrology, e.g., daily streamflow forecasting (Rasouli et al. 2012), rainfall–runoff modeling (Taormina & Chau 2015). Recently, Abdullah et al. (2015) and Feng et al. (2016) found ELM had very good generalization performances for ET0 estimation. In this study, we applied ELM for maize ET estimation, and different combinations of meteorological and crop were selected as input to train ELM. Figure 1 presents the structure of the ELM model, which consists of input layer (input variables), hidden layer (neurons), and output layer (ET). Different hidden nodes were used to estimate maize ET (e.g., 10, 20, 30, … , 200), and we found that with the increase of hidden nodes, the error of the ELM model reduced significantly and this reduction was not so significant after hidden nodes exceeded 100. Thus, 100 hidden nodes proved efficient to estimate maize ET. Further details about ELM may be found in Huang et al. (2006), Abdullah et al. (2015), and Feng et al. (2016).
Figure 1

The ELM structure.

Figure 1

The ELM structure.

Generalized regression neural network

The GRNN was first proposed by Specht (1991) and is a radial basis function network. GRNN approximates any arbitrary function between input and output vectors, drawing the function estimate directly from training data. Although it is similar to the common FFNN, its operation is fundamentally different in that it is based on nonlinear regression theory for function estimation. The training set consists of values of inputs x and produces the estimated value of y, which minimizes the squared error (Kisi 2006, 2008). The GRNN structure is presented in Figure 2. As shown in Figure 2, the GRNN consists of four layers, including the input layer (input variables), pattern layer, summation layer, and output layer (ET). The spread constants for the GRNN model are determined by using circuit training. Further details about GRNN for ET estimation can be found in Kisi (2006, 2008) and Ladlani et al. (2012). MATLAB software (R2015b) is utilized to implement ELM and GRNN for daily maize ET estimation.
Figure 2

The GRNN structure.

Figure 2

The GRNN structure.

Model training and assessment

Two different input combinations were selected to train the GRNN and ELM models. One considered meteorological and crop data as input, the same input data as the modified dual Kc approach, and the other combination considered meteorological data as input to evaluate the capabilities of the ELM and GRNN when only meteorological data are available. Table 1 presents a summary of the input combinations for each model. ELM1 and GRNN1 were fed with meteorological (maximum, minimum, and mean air temperature, maximum, minimum, and mean relative humidity, solar radiation, and wind speed at 2 m height) and crop data (leaf area index and plant height). ELM2 and GRNN2 were fed only with meteorological data since the crop data are not commonly available.

Table 1

Summary of input combinations for each AI model

  Meteorological data
 
Crop data
 
Model T RH Rs u2 LAI hc 
ELM1 √ √ √ √ √ √ 
GRNN1 √ √ √ √ √ √ 
ELM2 √ √ √ √   
GRNN2 √ √ √ √   
  Meteorological data
 
Crop data
 
Model T RH Rs u2 LAI hc 
ELM1 √ √ √ √ √ √ 
GRNN1 √ √ √ √ √ √ 
ELM2 √ √ √ √   
GRNN2 √ √ √ √   

Although k-fold assessment is recommended for assessing the performances of AI models (Martí et al. 2011b, 2015; Shiri et al. 2014a, 2014b, 2015), a simple data set assignment was considered for this study. The k-fold assessment would have involved computational costs that could not be assumed. The considered assessment procedure is a very common practice for AI models' assessment (Shiri et al. 2014c). The data of 2011 and 2012 were used to train the ELM and GRNN models, and data of 2013 were applied to assess the performances of the models. Thus, 293 patterns were available for training, while 150 patterns were used for testing.

T is air temperature, including maximum, minimum, and mean air temperature; RH is relative humidity, including maximum, minimum, and mean relative humidity; Rs is solar radiation; u2 is wind speed at 2 m height; LAI is leaf area index; hc is plant height.

Performance evaluation

Root mean square error (RMSE), mean absolute error (MAE), and Nash–Sutcliffe coefficient (NS) were used to evaluate performances of the models: 
formula
18
 
formula
19
 
formula
20
where and are ET values at the i-th step obtained by measurement and estimation, respectively; n is the number of the time steps; ETmean is the mean value of the measured ET values; RMSE and MAE are both in mm/d, taking on value from 0 (perfect fit) to ∞ (the worst fit); NS is dimensionless, taking on value from 1 (perfect fit) to −∞ (the worst fit).

RESULTS AND DISCUSSION

Variations of meteorological and crop variables

Figure 3 presents seasonal variations of meteorological variables during maize growing seasons of 2011–2013. Daily air temperature, including maximum, minimum, and mean air temperature, increased in initial and development stage, but decreased gradually in the mid and late stage, with average maximum, minimum, and mean air temperature of 23.5, 12.5, and 17.4 °C for 2011, 23.8, 13.5, 18.2 °C for 2012 and 25.2, 13.5, and 18.4 °C for 2013, respectively. Compared with minimum and mean relative humility, maximum relative humility had quite a different change pattern. It was lower at the initial and late stage, and maintained stable values above 90% due to heavier rainfall in the mid stage for 3 years, with average maximum, minimum, and mean relative humidity of 83.5%, 42.0%, and 66.9% for 2011, 84.5%, 42.8%, 67.7% for 2012 and 83.3%, 41.2%, and 66.5% for 2013, respectively. Solar radiation fluctuated significantly during the growing season for 3 years, and lower values can be observed in the initial and late stage with greater values in the development and mid stage, with average solar radiation of 18.6, 15.9, and 18.2 MJ/m2 d for 3 years, respectively. An obvious declining trend of wind speed from initial to late stage could be found, with an average wind speed of 2.0, 2.0, and 1.8 m/s for 3 years, respectively. ET0 increased in the initial stage and then decreased gradually to the late stage, with average ET0 of 3.9, 3.9, and 4.2 mm/d. Precipitation fluctuated significantly during the growing season for 3 years, and total precipitation of 2011 was lower than that of 2013 and greater than that of 2012, with total precipitation of 496, 417, and 515 mm for 3 years, respectively.
Figure 3

Seasonal variations of meteorological variables during maize growing seasons of 2011–2013.

Figure 3

Seasonal variations of meteorological variables during maize growing seasons of 2011–2013.

Figure 4 presents seasonal variations of LAI and hc for 3 years. LAI increased significantly after the initial stage, and reached a maximum in mid stage while decreasing in the late stage for all 3 years. Greater LAI can be observed in 2012, and it was about 2 weeks earlier than that of 2013 and 3 weeks earlier than that of 2011 when LAI reached the maximum, with LAI ranges of 0–3.64 m2/m2 for 2011, 0–4.52 m2/m2 for 2012, and 0–3.97 m2/m2 for 2013, respectively. Similarly, significant increases can be observed after the initial stage, and reached a maximum in mid stage. Greater hc can be observed in 2012, with hc ranges of 0–2.75 m for 2011, 0–2.98 m for 2012, and 0–2.97 m for 2013, respectively. Greater LAI and hc in 2012 may be attributed to greater soil temperature and soil water content in 2012.
Figure 4

Seasonal variations of leaf area index and plant height during maize growing seasons of 2011–2013.

Figure 4

Seasonal variations of leaf area index and plant height during maize growing seasons of 2011–2013.

Energy balance closure of the eddy covariance system

The relationships between available energy (Rn-G) and turbulent fluxes (LE + H) at interval of 30 minutes for 3 years are presented in Figure 5. It is seen that the slopes of the linear regression varied from 0.78 to 0.92, with average energy balance closure of 0.86, which is within common results found in the literature for eddy covariance systems (Li et al. 2005; Allen et al. 2011). In accordance with the findings of other studies (Wilson et al. 2002; Wolf et al. 2008; Allen et al. 2011; Zhang et al. 2013), imbalance of the eddy covariance system in the experimental site was found, which may be attributed to the underestimation of ET (Wolf et al. 2008).
Figure 5

Relationships between Rn-G and LE + H at intervals of 30 minutes for 3 years.

Figure 5

Relationships between Rn-G and LE + H at intervals of 30 minutes for 3 years.

Parameters of dual Kc approach

Table 2 presents parameters of dual Kc approach for maize ET estimation in 2013. Ze was 0.1 m in this study, recommended by FAO-56 (Allen et al. 1998). Measured maximum hc and LAI were 2.97 m and 3.95 m2/m2, respectively. REW, TEW, and TAW were calibrated using soil property data, with values of 7, 23, and 181 mm, respectively. Greater TEW and TAW may be due to the fact that the soil in the experimental site was sandy loam, leading to a greater soil water content at field capacity (36.2%). According to Allen & Pereira (2009) and Ding et al. (2013), accurate estimation of canopy cover coefficient could be achieved when κ was 0.7.

Table 2

Parameters of dual Kc approach for maize ET estimation in 2013

Parameters Values Units Source 
Ze 0.1 Allen et al. (1998)  
Maximum hc 2.97 Measured 
Maximum LAI 3.95 m2/m2 Measured 
REW mm Calibrated 
TEW 23 mm Calibrated 
TAW 181 mm Calibrated 
κ 0.7 – Ding et al. (2013)  
Parameters Values Units Source 
Ze 0.1 Allen et al. (1998)  
Maximum hc 2.97 Measured 
Maximum LAI 3.95 m2/m2 Measured 
REW mm Calibrated 
TEW 23 mm Calibrated 
TAW 181 mm Calibrated 
κ 0.7 – Ding et al. (2013)  

Figure 6 presents seasonal variation of crop coefficient in 2013. The crop coefficient increased in the initial and development stage, and reached maximum in the mid stage, then decreased gradually in the late stage, with crop coefficient ranges of 0.06–0.26 in the initial stage, 0.15–0.82 in the development stage, 0.67–1.43 in the mid stage, and 0.54–1.22 in the late stage.
Figure 6

Seasonal variation of crop coefficient during maize growing season of 2013.

Figure 6

Seasonal variation of crop coefficient during maize growing season of 2013.

Comparison of AI and dual Kc approaches

The comparison of the ET values estimated by the ELM, GRNN, and FAO-56 dual Kc models is shown in Figure 7, in the form of line graphs and scatter plots. It is seen from the line graphs and scatter plots that ET values estimated by the ELM1 model closely followed the measured ET values, and the slope of linear regression between estimated and measured ET was 0.974, with R2 of 0.998; ET values estimated by the GRNN1 model closely followed the measured ET values too, and the slope of linear regression was 0.968, with R2 of 0.997; the ELM2 and GRNN2 models, with only meteorological inputs, demonstrated poorer performances than ELM1 and GRNN1, with the slope of linear regression of 0.912 (R2 = 0.955) and 0.804 (R2 = 0.93), respectively, indicating LAI and hc were very influential parameters for ET estimation. The modified dual Kc approach showed a slightly poorer performance than those of the ELM1 and GRNN1 models, with the slope of linear regression of 0.928 (R2 = 0.976). Obvious underestimation of ET can be observed in the initial stage for the ELM1 model while GRNN1, ELM2, GRNN2, and dual Kc models overestimated ET in the initial stage. Compared with FAO-56 dual Kc approach, ELM1 and GRNN1 had higher goodness-of-fit, which confirmed the capabilities of ELM1 and GRNN1 for ET estimation.
Figure 7

Comparison of ET estimated by ELM, GRNN, and FAO-56 dual Kc models with the measured ones during maize growing seasons of 2013.

Figure 7

Comparison of ET estimated by ELM, GRNN, and FAO-56 dual Kc models with the measured ones during maize growing seasons of 2013.

Statistical performances of ELM, GRNN, and dual Kc models for maize ET estimation in 2013 are presented in Table 3. Based on the statistical indicators, ELM1 had the best performances for ET estimation, with RMSE of 0.221 mm/d, MAE of 0.203 mm/d, and NS of 0.981, respectively; estimated ET by ELM1 was 364.8 mm, which was 1.3% lower than measured ET. GRNN1 had good performances for ET estimation, too, with RMSE of 0.225 mm/d, MAE of 0.211 mm/d, and NS of 0.981, respectively; estimated ET by GRNN1 was 378.3 mm, which was 2.4% greater than measured ET. The performances of FAO-56 dual Kc approach were poorer than those of ELM1 and GRNN1, but better than those of ELM2 and GRNN2, with RMSE of 0.381 mm/d, MAE of 0.332 mm/d, and NS of 0.871, respectively. Although ELM2 and GRNN2 were not as efficient as ELM1, GRNN1, and dual Kc models, their estimation of ET was acceptable when only meteorological data were available.

Table 3

Statistical performances of ELM, GRNN, and dual Kc models for maize ET estimation in 2013

Model Estimated ET (mm) Over/Underestimation (%) RMSE (mm/d) MAE (mm/d) NS 
ELM1 364.8 −1.3 0.221 0.203 0.981 
GRNN1 378.3 2.4 0.225 0.211 0.981 
ELM2 398.6 7.9 0.403 0.353 0.848 
GRNN2 400.8 8.5 0.521 0.421 0.836 
FAO-56 385.6 4.4 0.381 0.332 0.871 
Model Estimated ET (mm) Over/Underestimation (%) RMSE (mm/d) MAE (mm/d) NS 
ELM1 364.8 −1.3 0.221 0.203 0.981 
GRNN1 378.3 2.4 0.225 0.211 0.981 
ELM2 398.6 7.9 0.403 0.353 0.848 
GRNN2 400.8 8.5 0.521 0.421 0.836 
FAO-56 385.6 4.4 0.381 0.332 0.871 

Overall, this study found that the ELM and GRNN models can be applied successfully for maize ET estimation. Compared with GRNN, ELM was more efficient. In contrast to traditional FFNN, ELM randomly chooses hidden nodes and analytically determines the output weights, which may result in better performances of ELM. Although good performances of ELM and GRNN were found, they have no physical basis and belong to a class of data-driven black-box approaches (Tabari et al. 2012, 2013). In addition, they cannot partition ET separately into evaporation and transpiration.

CONCLUSION

The potential of ELM and GRNN for estimation of rainfed maize evapotranspiration was investigated on the China Loess Plateau in this study. A field experiment was conducted during maize growing seasons of 2011–2013 for continuous measurements of maize ET with eddy covariance systems, meteorological variables with automatic weather station, LAI, and hc. These data were used to train the ELM and GRNN models consisting of two combinations of meteorological and crop parameters. The ELM1 and GRNN1 models whose inputs were meteorological and crop data performed better than the modified dual Kc model, which confirmed the capabilities of ELM and GRNN models for maize ET estimation. Although the ELM2 and GRNN2 models using only meteorological data were not as efficient as ELM1, GRNN1, and dual Kc models, their accuracy for maize ET estimation was acceptable, and could be considered as a tool to estimate maize ET when crop data are insufficient.

ACKNOWLEDGEMENTS

We are grateful for the research grants from the National Natural Science Foundation of China (No. 51179194), National Key Technologies R&D Program of China (No. 2015BAD24B01, No. 2012BAD09B01) and Basic Science Research Foundation of China Central Government (BSRF201609). Cordial thanks are extended to the editor and three anonymous reviewers for their valuable comments.

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