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 (*h _{c}*) were continuously measured during maize growing seasons of 2011–2013. The meteorological data and crop data including

*LAI*and

*h*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 (

_{c}*K*) approach in 2013. Results indicated that ELM1 and GRNN1 using meteorological and crop data as inputs estimated maize

_{c}*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

*K*model. Performances of ELM2 and GRNN2 using only meteorological data as input were poorer than those of ELM1, GRNN1, and modified dual

_{c}*K*approach, but its estimation of maize

_{c}*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 (*ET _{0}*) and crop coefficient (

*K*

_{c}).

*ET*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

_{0}*K*

_{c}, the ratio of

*ET*and

*ET*, represents the effects of characteristics that distinguish the specific field crops from the reference crop. Compared with the single

_{0}*K*

_{c}method, the dual

*K*

_{c}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

*K*

_{c}approach is needed when applying the method for

*ET*estimation.

In recent years, artificial intelligence (AI) has been successfully implemented in *ET _{0}* estimation. The application of AI in

*ET*modeling was first investigated after 2000 by Kumar

_{0}*et al.*(2002), who estimated

*ET*using an artificial neural network (ANN). Since then, efforts towards the estimation of

_{0}*ET*using ANN models have been performed (Trajkovic

_{0}*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

*ET*estimation: Tabari

_{0}*et al.*(2012) investigated the performances of support vector machines, adaptive neuro-fuzzy inference system for

*ET*estimation in a semi-arid highland environment in Iran; Shiri

_{0}*et al.*(2011, 2012) applied a genetic programming approach for

*ET*and evaporation modeling; Kisi

_{0}*et al.*(2012) developed generalized neurofuzzy-based evaporation models in Arizona, USA; Pour Ali Baba

*et al.*(2013) estimated

*ET*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

_{0}*ET*in Turkey; Shiri

_{0}*et al.*(2013) and Martí

*et al.*(2015) used gene expression programming for

*ET*estimation; Kisi (2016) investigated the ability of least square support vector regression, multivariate adaptive regression splines and M5 Model Tree in modeling

_{0}*ET*in Turkey; Abdullah

_{0}*et al.*(2015) and Feng

*et al.*(2016) examined the capability of extreme learning machine (ELM) for

*ET*estimation in Iraq and Southwest China, respectively. All these studies confirmed good performances of AI approaches for

_{0}*ET*estimation worldwide. Differently to

_{0}*ET*, which is from a reference crop and only affected by meteorological variables,

_{0}*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 *K*_{c} 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/cm^{3}. 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 (*R*_{s}) 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 (*u*_{2}) 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

*h*), 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 (

_{c}*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): where

*LAI*is leaf area index (m

^{2}/m

^{2});

*L*and

_{i}*W*is the length and width of the

_{i}*i*-th leaf, respectively;

*D*and

_{row}*S*stand for the distance between the two rows and the space between the plants in the row, respectively.

_{plant}### The modified dual *K*_{c} approach for *ET* estimation

#### Reference evapotranspiration

*ET*) was estimated using the FAO-56 Penman–Monteith equation (Allen

_{0}*et al.*1998; Feng

*et al.*2014): where

*ET*is reference evapotranspiration (mm/d);

_{0}*R*is net radiation (MJ/m

_{n}^{2}d);

*G*is soil heat ﬂux density (MJ/m

^{2}d);

*T*is mean air temperature (°C);

_{mean}*e*is saturation vapor pressure, kPa;

_{s}*e*is actual vapor pressure, kPa;

_{a}*Δ*is slope of the saturation vapor pressure function (kPa/°C);

*γ*is psychometric constant (kPa/°C); and

*u*

_{2}is wind speed at 2 m height (m/s).

#### Maize ET

*ET*was calculated by multiplying

*ET*and

_{0}*K*

_{c}(Allen

*et al.*1998): According to the dual

*K*approach,

_{c}*K*can be split into two parameters, which are

_{c}*K*and

_{e}*K*. Thus, Equation (3) can be rewritten as: where

_{cb}*ET*is crop evapotranspiration (mm/d);

*K*is basal crop coefficient;

_{cb}*K*is soil evaporation coefficient;

_{e}*K*is water stress coefficient. Thus,

_{s}*K*represents crop transpiration while

_{s}K_{cb}ET_{0}*K*represents soil evaporation.

_{e}ET_{0}(1) Calculation of

*K*:_{cb}

*K*is calculated as (Allen

_{cb}*et al.*1998): where

*K*is the original basal crop coefficient obtained;

_{cb,o}*K*

_{cb}_{(Tab)}is basal crop coefficient value taken from FAO-56 recommendation;

*u*is wind speed at 2 m height (m/s);

_{2}*RH*is minimum daily relative humidity (%); and

_{min}*h*is crop height (m).

_{c}*K*) to accurately calculate daily actual

_{cc}*K*(Ding

_{cb}*et al.*2013): where

*K*is the minimum basal crop coefﬁcient for bare soil;

_{c,min}*K*is basal crop coefﬁcient when crop has nearly full ground cover.

_{cb,full}*K*can be calculated following the method of Allen

_{cb,full}*et al.*(1998):

*K*was then calculated by the ratio of radiation intercepted by crop canopy: where

_{cc}*κ*is the coefficient for radiation extinguish and

*LAI*is leaf area index.

(2) Calculation of

*K*_{e}:

*K*

_{e}is calculated by estimating energy availability and soil moisture regimes at the soil surface through the original FAO-56 procedure: where

*K*is the soil evaporation coefﬁcient;

_{e}*K*is the maximum value of

_{c,max}*K*following rain or irrigation;

_{c}*K*is evaporation reduction coefﬁcient;

_{r}*f*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.,

_{ew}*K*could not exceed few

_{e}*K*.

_{c,max}*K*is adjusted for crop height and climate: where

_{c,max}*f*

_{c}is the approximate fraction of soil surface that is exposed;

*f*is the average fraction of soil surface wetted by irrigation or precipitation;

_{w}*D*

_{e,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

*D*≤

_{e,i-1}*REW*,

*K*

_{r}= 1.0. where

*θ*and

_{FC}*θ*are soil water contents at ﬁeld capacity and wilting point, respectively (m

_{WP}^{3}/m

^{3}).

*Z*is depth of the topsoil layer.

_{e}*f*was determined by

_{c}*LAI*:

*D*calculation requires a daily water budget for the exposed and wetted surface. where

_{e}*D*and

_{e,i}*D*are cumulative depth of evaporation of the topsoil at the end of day

_{e,i-1}*i*and

*i*−1 (mm);

*P*is precipitation on day

_{i}*i*(mm);

*RO*is runoff from the soil surface on day

_{i}*i*(mm);

*E*is evaporation on day

_{i}*i*(mm);

*T*is depth of transpiration of the soil surface layer on day

_{ew,i}*i*(mm),

*T*= 0 for row crops as recommended in Allen

_{ew,i}*et al.*(1998);

*DP*is deep percolation loss from the topsoil layer on day

_{e,i}*i*if soil water content exceeds ﬁeld capacity (mm).

*D*

_{r}_{,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

*D*

_{r}_{,i-1}<

*RAW*,

*K*= 1.0. where

_{s}*Z*is the rooting depth (m).

_{r}### AI models for *ET* estimation

#### Extreme learning machine

*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

*ET*estimation. In this study, we applied ELM for maize

_{0}*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).

#### Generalized regression neural network

*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.

#### 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 *K _{c}* 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.

Meteorological data | Crop data | |||||
---|---|---|---|---|---|---|

Model | T | RH | R _{s} | u _{2} | LAI | h _{c} |

ELM1 | √ | √ | √ | √ | √ | √ |

GRNN1 | √ | √ | √ | √ | √ | √ |

ELM2 | √ | √ | √ | √ | ||

GRNN2 | √ | √ | √ | √ |

Meteorological data | Crop data | |||||
---|---|---|---|---|---|---|

Model | T | RH | R _{s} | u _{2} | LAI | h _{c} |

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; *R _{s}* is solar radiation;

*u*is wind speed at 2 m height;

_{2}*LAI*is leaf area index;

*h*is plant height.

_{c}### Performance evaluation

*ET*values at the

*i*-th step obtained by measurement and estimation, respectively;

*n*is the number of the time steps;

*ET*is the mean value of the measured

_{mean}*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

^{2}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.

*ET*increased in the initial stage and then decreased gradually to the late stage, with average

_{0}*ET*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.

_{0}*LAI*and

*h*for 3 years.

_{c}*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 m

^{2}/m

^{2}for 2011, 0–4.52 m

^{2}/m

^{2}for 2012, and 0–3.97 m

^{2}/m

^{2}for 2013, respectively. Similarly, significant increases can be observed after the initial stage, and reached a maximum in mid stage. Greater

*h*can be observed in 2012, with

_{c}*h*ranges of 0–2.75 m for 2011, 0–2.98 m for 2012, and 0–2.97 m for 2013, respectively. Greater

_{c}*LAI*and

*h*

_{c}in 2012 may be attributed to greater soil temperature and soil water content in 2012.

### Energy balance closure of the eddy covariance system

*R*-

_{n}*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).

### Parameters of dual *K*_{c} approach

_{c}

Table 2 presents parameters of dual *K _{c}* approach for maize

*ET*estimation in 2013.

*Z*was 0.1 m in this study, recommended by FAO-56 (Allen

_{e}*et al.*1998). Measured maximum

*h*and

_{c}*LAI*were 2.97 m and 3.95 m

^{2}/m

^{2}, 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.

Parameters | Values | Units | Source |
---|---|---|---|

Z _{e} | 0.1 | m | Allen et al. (1998) |

Maximum h _{c} | 2.97 | m | Measured |

Maximum LAI | 3.95 | m^{2}/m^{2} | Measured |

REW | 7 | mm | Calibrated |

TEW | 23 | mm | Calibrated |

TAW | 181 | mm | Calibrated |

κ | 0.7 | – | Ding et al. (2013) |

Parameters | Values | Units | Source |
---|---|---|---|

Z _{e} | 0.1 | m | Allen et al. (1998) |

Maximum h _{c} | 2.97 | m | Measured |

Maximum LAI | 3.95 | m^{2}/m^{2} | Measured |

REW | 7 | mm | Calibrated |

TEW | 23 | mm | Calibrated |

TAW | 181 | mm | Calibrated |

κ | 0.7 | – | Ding et al. (2013) |

### Comparison of AI and dual *K*_{c} approaches

*ET*values estimated by the ELM, GRNN, and FAO-56 dual

*K*

_{c}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

*R*of 0.998;

^{2}*ET*values estimated by the GRNN1 model closely followed the measured

*ET*values too, and the slope of linear regression was 0.968, with

*R*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 (

^{2}*R*= 0.955) and 0.804 (

^{2}*R*= 0.93), respectively, indicating

^{2}*LAI*and

*h*were very influential parameters for

_{c}*ET*estimation. The modified dual

*K*

_{c}approach showed a slightly poorer performance than those of the ELM1 and GRNN1 models, with the slope of linear regression of 0.928 (

*R*= 0.976). Obvious underestimation of

^{2}*ET*can be observed in the initial stage for the ELM1 model while GRNN1, ELM2, GRNN2, and dual

*K*models overestimated

_{c}*ET*in the initial stage. Compared with FAO-56 dual

*K*approach, ELM1 and GRNN1 had higher goodness-of-fit, which confirmed the capabilities of ELM1 and GRNN1 for

_{c}*ET*estimation.

Statistical performances of ELM, GRNN, and dual *K*_{c} 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 *K _{c}* 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

*K*models, their estimation of

_{c}*ET*was acceptable when only meteorological data were available.

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 *h _{c}*. 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

*K*model, which confirmed the capabilities of ELM and GRNN models for maize

_{c}*ET*estimation. Although the ELM2 and GRNN2 models using only meteorological data were not as efficient as ELM1, GRNN1, and dual

*K*models, their accuracy for maize

_{c}*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.