Support vector machine based modeling of evapotranspiration using hydro-climatic variables in a sub-tropical environment

Agricultural and Forest Meteorology 200 (2015) 172–184

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Support vector machine based modeling of evapotranspiration using hydro-climatic variables in a sub-tropical environment N.K. Shrestha, S. Shukla ∗ Department of Agricultural and Biological Engineering, University of Florida, 2685 State Road 29N, Immokalee, FL 34142

a r t i c l e

i n f o

Article history: Received 7 November 2013 Received in revised form 25 July 2014 Accepted 27 September 2014 Keywords: Evapotranspiration Crop coefficients Modeling Prediction SVM

a b s t r a c t Existing models and methods report crop coefficient (Kc ) as a function of time but do not consider the variations due to surface conditions, wetting methods, meteorological conditions, and other biophysical factors. These limitations result in erroneous crop evapotranspiration (ETc ) estimates, especially for nonstandard conditions (e.g. plastic mulch). We present Support Vector Machine (SVM), a data-driven model based on statistical learning theory, for predicting generic Kc and ETc using a uniquely large dataset (10 seasons) from lysimeters for multiple crop-seasons combination under the plastic mulch conditions. The data used in this study were obtained from six years of lysimeter-based measurements (Shukla et al., 2006, 2012, 2014a, 2014b; Shukla and Knowles, 2011) for two distinctly different crop types (vine and erect) under two contrasting irrigation methods, drip and sub-irrigation. The SVM-based models predicted bell pepper (erect crop; r2 = 0.71) and watermelon (vine crop; r2 = 0.82) Kc as a function of time, water table depth, and number of rainfall events. The time since transplant represents the plant growth and, therefore, transpiration. The water table depth and rainfall events capture the effect of surface soil moisture on evaporation. The crop type-specific model is robust since it works for two different irrigation methods and growing seasons (spring and fall). The SVM model was superior to the Artificial Neural Network and Relevance Vector Machine models, two data-driven models used in hydrology. The errors in predicting ETc from the SVM model were only 2.6% and 11.2% for watermelon and bell pepper, respectively, highlighting the model accuracy. For both crops, the SVM predicted Kc values were not statistically different from the actual Kc values. In contrast, the FAO-56 values were significantly lower than the actual Kc values for both bell pepper (p = 0.016) and watermelon (p = 0.025). When evaluated in the context of watershed-scale budgets, the SVM model improved the accuracy in ETc estimates by 49.3 mm over the FAO-56 method, and this improvement represents 70% (70.7 mm) of the observed surface flow. Improved accuracy of the SVM model makes it useful in deriving local Kc using readily available hydro-climatic data for applications ranging from field-scale water management to watershedscale modeling. The proposed model can be used to develop region-specific Kc to improve ETc estimates. Future efforts should be made to explore the development of similar models for open-field crops. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The crop coefficient (Kc ) approach has been one of the most frequently used methods for estimating ETc for the last 40 years (Allen et al., 1998; Tasumi et al., 2005). It has been used for the applications ranging from irrigation scheduling (Allen et al., 1998), simulating watershed-scale water balance components (Jaber and Shukla, 2012) to daily flux modeling (Payero and Irmak, 2013). Crop coefficient (Kc ) is a ratio of crop evapotranspiration (ETc ) to reference

∗ Corresponding author. Tel.: +1 239 658 3425 E-mail address: sshukla@ufl.edu (S. Shukla). http://dx.doi.org/10.1016/j.agrformet.2014.09.025 0168-1923/© 2014 Elsevier B.V. All rights reserved.

evapotranspiration (ET0 ). Literature Kc values (Doorenbos and Pruitt, 1977; Wright, 1981; Allen et al., 1998) are specific to crop type, climatic region, and type of soil wetting. The Kc developed for a specific production system and climatic region is likely to be different from other production systems and regions (Allen et al., 1998; Liu et al., 2002; Kang et al., 2003). The difference between literature and locally-developed Kc values has been reported for both agricultural crops (Simon et al., 1998; Kashyap and Panda, 2001; Kang et al., 2003; Lovelli et al., 2005; Orgaz et al., 2005; Shukla et al., 2012, 2014a, 2014b) and wetland vegetations (Wu and Shukla, 2013), and is to be expected. The differences become especially important when the surface cover condition is different (e.g. plastic mulch) from the standard open-field condition.

N.K. Shrestha, S. Shukla / Agricultural and Forest Meteorology 200 (2015) 172–184

Crop production under plastic mulch has become a common production system for growing vegetable (e.g. pepper) and fruit (e.g. melons) crops worldwide including Florida. Covering raised beds with impermeable plastic mulch heats the soil through a greenhouse effect (Shinde et al., 2001). These effects combined with the additional benefit of reducing evaporation losses, nutrient leaching, and disease pressure and preventing weed competition make plasticulture an increasingly attractive production system. Almost 12 million hectares of plastic mulch covered farmland existed worldwide in 1999, a number which most likely has increased significantly by now (Miles et al., 2005). The water requirement of the crops grown with plasticulture is different than that in an open-field system, as it significantly alters the water and energy fluxes. Plastic mulch prevents rainfall entry and reduces overall evaporation but increases transpiration. Mulch increases the water flux and infiltration in the row-middle areas because of runoff from the raised beds. Allen et al. (1998) recommended reduction in Kc by 10–30% depending on wetting interval for the drip irrigation system under the plastic mulch. Studies show a difference in Kc values compared to FAO-56 for crops grown under plastic mulch (Amayreh and Al-Abed, 2005; Lovelli et al., 2005; Bryla et al., 2010; Moratiel and Martinez-Cob, 2012; Shukla et al., 2014a, 2014b). In Florida and elsewhere where plasticulture is practiced, wetting of soil prior to bed formation results in soil moisture to near saturation. Further wetting of the soil by rainfall and upflux from a shallow water table at the beginning of growing season keeps the soil moisture high in the row-middle areas, increasing evaporation, which can be 30–60% of the seasonal crop ETc (Liu et al., 2002; Agam et al., 2012). The increased evaporation can significantly affect ETc and Kc . With the current models and published methods, an accurate estimate of ETc for the crops grown under plastic mulch remains a challenge. Depending on the irrigation methods and climate (Allen et al., 2005; Zhou and Zhou, 2009), Kc and ETc can vary for the same crop. Sub-irrigation and drip irrigation systems are two common irrigation methods in Florida. The sub-irrigation method in Florida involves surface application of water into ditches similar to surface irrigation. The water is applied at a high rate on the surface, which artificially raises the water table within 0.6 m from the surface. Although sub-irrigation has relatively low efficiency (50%; Smajstrla et al., 1991) compared to drip irrigation (>80%, Lamm and Trooien, 2003), its low cost makes it an attractive option for farmers. Considerable volume of water is lost through evaporation for flood irrigation compared to drip (Lazzara and Rana, 2010), where flood and sub-irrigation have similar efficiency. Drip irrigation under plastic mulch increases its efficiency further because it reduces unproductive evaporation. Existing models and methods lack the robustness and accuracy to differentiate between drip and subirrigated crops to calculate ET fluxes. The uncertainty associated with the Kc approach arising from literature Kc values that do not represent the climate and production method of the system being simulated can result in poor calibration and validation of hydrologic models (Kienzle and Schmidt, 2008; Zhao et al., 2012), and often leads to transferring the errors to other water balance components. The differences need to be quantified, which necessitates the development of a model to estimate Kc for the type of wetting for a plasticulture production system. For a plasticulture system, the soil is wetted to near saturation for preparing the firm raised beds with a tractor-driven machine. Due to excessive wetting, the evaporation is high compared to transpiration at the initial crop stage for both sub- and drip-irrigation methods as the soil is wetted to the same level. When the soil moisture is not limiting, the evaporation rate is close to ET0 during the early crop stage (Liu et al., 2002). The evaporation is also affected by the crop cover or structure. The effective full cover for an erect crop such as bell pepper is about 40% (Miranda et al., 2006), which

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results in high evaporation from the wide-open bare row-middles. The vine crop such as watermelon affects the row-middle evaporation differently than an erect crop. Evapotranspiration for a vine crop during the early crop growth stage predominantly occurs in the form of soil evaporation when the crop cover is low. As the crop grows, greater shading from its canopy reduces bare soil evaporation (Johnson, 2002). Ground cover for a vine crop is both larger and spreads faster than that for an erect crop. Most vine crops achieve almost complete ground cover (>95%; Akintoye et al., 2009) at full growth, resulting in diminished soil evaporation and increased transpiration. During this stage, wetting of row-middles by rainfall and/or irrigation does not have much effect on evaporation. Evaporation of rain or irrigation from a vine crop, therefore, is likely to be different from an erect low canopy crop due to differences in bare soil evaporation as well as direct evaporation of the intercepted rainfall. In addition to fraction of ground cover (Williams and Ayars, 2005), the crop height also affects Kc (Allen and Pereira, 2009). Since vine and erect crops are distinctly different in crop height and ground cover, significant difference in ETc and Kc between the two types of crop is expected. Many regions in the world including Florida have multiple cropping seasons (e.g. gradually warming, gradually cooling) where the same crop is grown for more than one season within a year that are distinctly different in climatic conditions. The seasonal difference in climate variables and crop growth (Went, 1953) can result in a difference in ETc and Kc between the seasons. In absence of local Kc values, same literature Kc values have been used for the meteorologically different seasons despite the fact that they are mostly derived under specific climatic conditions. This adds further uncertainty in crop water use estimates. Past studies report Kc as a function of time, but do not consider the variation in Kc for the factors such as hydrologic conditions, weather, wetting methods (e.g. drip, sub-irrigation), and surface conditions (e.g. plastic mulch, open-field, partial or complete cover). The majority of literature Kc values have been derived from lysimeter studies (Allen et al., 1998; Ko et al., 2009; Shukla et al., 2012, 2014a, 2014b) and are specific to crop, irrigation method, surface condition, and climatic region and season. The effects of such variables on ETc and Kc values have been reported. Doorenboss and Pruitt (1977) emphasized the need for local calibration of Kc under given climatic conditions. Allen et al. (1998) proposed adjustment on mid and late stage Kc for relative humidity, wind speed, and crop height. Zhou and Zhou (2009) performed regression analysis for modeling ETc for reed marsh in northern China, and found air temperature, relative humidity, and net radiation as the most explanatory variables for explaining variations of Kc . They also observed increased ETc after rain events. Li et al. (2003) conducted lysimeter studies for maize and wheat in China where ETc was strongly affected by rainfall, irrigation, and leaf area index (LAI). Rijal et al. (2012) analyzed the effects of subsurface drainage on corn and soybean ETc and Kc in North Dakota and found that ETc values were mainly affected by the water table level. A similar effect of water table on ETc is reported in Nachabe et al. (2005). Therefore, there is a need to develop a model that accounts for hydrologic and meteorological conditions to predict Kc in order to reduce uncertainty in ETc estimates. Current science of ET is not mature enough to accurately simulate the effect of type of surface wetting, season, and hydrologic condition on ETc using physically-based models. ETc modeling using current physically based models use simplified assumptions and have difficulty and extensiveness of the parameter estimation (Farahani and Ahuja, 1996). The complexities in the physically based models (Beven, 1989; Kirchner, 2006) and difficulties associated with the data acquisitions such as lysimeter-based Kc development (Bryla et al., 2010; Shukla et al., 2014a) along with the associated high costs have also limited the development of

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physically-based models for estimating local Kc and ETc . To overcome the limitations of physically-based models and easing the local development of Kc using limited data that strongly affect ETc , a data-driven model based on statistical learning theory can be used as an alternative approach and is the focus of this study. The data-driven model based on machine learning approach can provide reasonable predictions using limited numbers of readily available data compared to the requirement of physically based models. They can be robust and can provide reasonable prediction of the system behavior (Khalil et al., 2006). Although a variety of data-driven models have been used, the amount of data needed to support the development of such models for predicting local Kc has not been available. Recently, this level of data has become available from the lysimeters in south Florida (Shukla et al., 2006, 2012, 2014a, 2014b; Shukla and Knowles, 2011) for the development of such models for predicting local Kc for crops with different canopy structures. Artificial Neural Network (ANN), Support Vector Machine (SVM), and Relevance Vector Machine (RVM) are data-driven models based on statistical learning theory. The ANN model has been successfully used in hydrologic modeling (e.g. rainfall-runoff modeling, streamflow predictions) (Halff et al., 1993; Smith and Eli, 1995; Markus et al., 1995; Hsu et al., 1997; Khalil et al., 2005) as well as to identify key environmental variables affecting fluxes of carbon dioxide, latent heat, and sensible heat (Schmidt et al., 2011). However, ANN has not yet been used for predicting actual ET (ETc ). The RVM model is based on sparse Bayesian learning and has been used for streamflow prediction (Ghosh and Mujumdar, 2008; Shrestha, 2014) and reservoir operation (Ticlavilca, 2010). The SVM model for regression problems (Vapnik, 1995) is an extension of the SVM model for classification problems developed by Vapnik and co-workers in the early 1990s (Cortes and Vapnik, 1995). The SVM model has the ability to capture the underlying physics of the system by relating inputs and outputs through robust mathematical equations. The SVM model has been successfully used for predicting watershed water balance components such as streamflow (Asefa et al., 2006; Kalra and Ahmad, 2009). The SVM model always finds a global optimum (Vapnik, 1995) compared to the ANN model which may converge to local optima (Yao, 1999) and is prone to overfitting making the SVM model superior. Superior performance of the SVM model compared to ANN has been observed in hydrology (Dibike et al., 2001; Gill et al., 2006). The RVM model uses fewer data points in the training phase than the SVM model (Tipping, 2001). For the large data set, use of sparse data in the training phase by the RVM model may reduce noise and may perform relatively better than the SVM model (Khalil et al., 2006), while it can be opposite for the small dataset as fewer data points in the RVM model may fail to capture the variability in the data. Use of machine learning models in predicting ETc has been limited. Few past studies have used machine learning models including the SVM model for estimating ET, however, they were used for predicting evaporation (E) and ET0 (Kumar et al., 2002; Torres et al., 2011; Samui and Dixon, 2012), both of which are easier to predict with relatively accurate models. While a variety of models exist, FAO-PM (Allen et al., 1998) is widely used worldwide as a standard and relatively accurate model for estimating ET0 . The main source of uncertainty is in predicting ETc , on which limited studies have been conducted. The use of machine learning models in ETc modeling has been limited to predictions at continental scale (Yang et al., 2006) which is not likely to be applicable for accurate estimates at the local (farm-scale) level. For example, 34% difference in ETc has been observed for the same crop grown at the same location but with drip and sub-irrigation systems (Shukla et al., 2014a). Differences in ETc due to different wetting methods are not likely to be captured with the continental-scale models.

Despite the success of the SVM model in predicting hydrologic components (Asefa et al., 2006; Kalra and Ahmad, 2009), the SVM models have not yet been developed to predict ETc (and Kc ) under diverse irrigation, growing season, and crop type for the mulched systems due mainly to the complexity of the problem and the lack of a sufficiently large data set needed for the model. If the SVM model is capable of predicting Kc from the readily available crop, climate, and hydrologic data, it could help in water conservation by improving the irrigation management (scheduling) and making accurate water allocation plans by improving the predictions of the hydrologic models. After validation, the SVM model can be used to predict local Kc rather than using the resource intensive methods such as lysimeter and eddy covariance methods (Allen et al., 2011). The objectives of this study are to: (1) develop a SVM-based model for predicting local Kc for two crop types with a wide range of crop cover (vine and erect), wetting method (drip and sub-irrigation), and season (gradually cooling fall and warming spring) using hydrologic and meteorological data in sub-tropical Florida and evaluate it against two other data-driven models, ANN and RVM; and (2) evaluate the accuracy of the SVM model by comparing its predictions with the measured data from the lysimeters and FAO-56 based Kc values. Watermelon is used here as an example for a vine crop while bell pepper (pepper) is an example for an erect crop. 2. Materials and methods 2.1. Study area The lysimeters for this study were installed at the Southwest Florida Research and Education Center (SWFREC), University of Florida/Institute of Food and Agricultural Science (UF/IFAS), Immokalee, Florida. The south Florida region has a sub-tropical climate with a naturally occurring shallow water table located at 0.6–0.8 m below the ground surface which can reach to the surface during the wet season (June–October). Soils in south Florida are highly sandy and hydric (Liudahl et al., 1998). Mean monthly maximum and minimum temperatures are 29 and 17 ◦ C, respectively and the annual rainfall is 1260 mm (1970–2009 data, Southeast Regional Climate Center, 2010). Most of the rainfall (about 70%) occurs during the wet season when row crops are not grown (Shukla et al., 2010). Both spring and fall seasons are dry and have low rainfall with the exception of two months (September and October) for the fall and, therefore, require supplemental irrigation to meet the crop water demand. 2.2. Lysimeter Data ETc and Kc values used in this study were derived from the measured water balance components from six large drainage lysimeters (4.85 × 3.65 × 1.35 m3 ) from Shukla et al. (2006, 2012, 2014a, 2014b; Shukla and Knowles, 2011). Four lysimeters were irrigated using drip method, while two were irrigated using a sub-irrigation (seepage) system. The water balance components of the lysimeters including irrigation, drainage, and runoff were measured with a propeller type flow meter connected to a datalogger (Shukla et al., 2006). The water table depth was measured using a pressure transducer installed in a monitoring well inside each lysimeter. The soil moisture data were measured using a capacitance-based soil moisture measurement device (Diviner 2000 and Enviroscan, Sentek Sensor Technologies, Australia) at two locations, one installed in the raised bed covered with plastic mulch and one in the row-middles. The measurements were conducted for 0.1 m depth increment from 0.0 to 0.7 m. Meteorological variables including air temperature, wind speed, relative humidity, solar radiation, and rainfall

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Table 1 Planting and harvesting dates for drip and sub-irrigated pepper and watermelon for different seasons. Year

Crop

2003a 2003 2004 2004 2005 2006 2006 2007 2007 2008 a

Season

Watermelon Pepper Watermelon Pepper Watermelon Pepper Pepper Pepper Pepper Pepper

Spring Fall Spring Fall Spring Spring fall Spring Fall Fall

Fall

Spring Planting

Harvesting

Planting

Harvesting

3-Apr – 8-Mar – 1-Mar 17-Feb – 14-Mar – -

30-May – 28-May – 31-May 25-May – 5-Jun – -

– 10–16 Sep – 16–Sep – – 12-Sep – 12–28 Sep 24-Sep

– 18–19 Dec – 22-Dec – – 21-Dec – 18–21 Dec 16-Dec

Experiment terminated after 8th week due to disease, - crops not grown.

were collected at the UF/IFAS Florida Automated Weather Network (FAWN) weather station located 50 m away from the lysimeters. The water balance components, meteorological data as well as ETc and Kc values for peppers were collected for two spring (March–May) 2006 and 2007, and five fall (September–December) 2003, 2004, 2006, 2007, and 2008 growing seasons for both drip and sub-irrigation systems. For watermelon, the data were collected for both irrigation systems for spring growing seasons of 2003, 2004, and 2005. Table 1 shows the growing periods (planting to harvest) for both crops. To compare Kc values from this study with literature (e.g., FAO-56), they were divided into four stages. Days after transplant (DAT) 0–20, 21–48, 49–83, and 84–97 were used as initial, development, mid-season, and late stages for pepper, while respective values for watermelon were DAT 0–13, 14–48, 49–69, and 70–83.

Table 2 Input sets for model development for pepper and watermelon. Input set

Variables

Input 1

a DATw,p , irrigation frequencyp , water table depthw,p , b soil moisturew,p , relative humidityw,p , rainfallw,p , c rainfall eventsw,p , solar radiationw,p , drainage and runoff frequencyp , average, minimum, maximum air temperaturew,p , average, minimum, and maximum wind speedw,p DATw,p , irrigation frequencyp , water table depthw,p , relative humidityw,p , rainfallw,p , rainfall eventsw,p , solar radiationw,p , drainage and runoff frequencyp , average, minimum, maximum air temperaturew,p , average, minimum, maximum wind speedw,p DATw,p , irrigation frequencyp , water table depthw,p , relative humidityw,p , rainfallw,p , rainfall eventsw,p , solar radiationw,p , drainage and runoff frequencyp , average air temperaturew,p , average wind speedw,p DATw,p , irrigation frequencyp , water table depthw,p , relative humidityw,p , rainfallw,p , rainfall eventsw,p , drainage and runoff frequencyp , average air temperaturew,p , solar radiationw DATw,p , irrigation frequencyp , water table depthw,p , rainfallw , rainfall eventsw,p , average air temperaturew,p DATw,p , water table depthw,p , rainfall frequencyp , average air temperaturew,p , rainfallw , relative humidityw DATw,p , water table depthw,p , rainfall eventsw,p , average air temperaturew DATw,p , water table depthw,p , average air temperaturew,p DATw , water table depthw , rainfall eventsw

Input 2

Input 3

Input 4

2.3. Support Vector Machine Input 5

For a given data set {(x1 , y1 ), ..................(xL , yL )} ⊂ X, where X denotes the space of the input patterns that consist of climate and hydrologic variables for a given crop type (Table 2), the goal is to find a functional dependency f(x) between inputs x and target y (represents Kc ) taken from the set of independent and identically distributed (i.i.d.) observations. Although i.i.d. is the assumption of machine learning models, for practical applications this assumption may not be realistic (Le Boudec, 2010). We tested the i.i.d. assumption and found it to be valid for our data set. The objective function for the SVM (Vapnik, 1995) is formulated through:

 1 ∗ Minimize w2 + C (i + i ) 2 L

i=1



Subject to,



yi − w, xi − b ≤ ε + i





w, xi + b − yi ≤ ε + i

(1)

∗

Input 6 Input 7 Input 8 Input 9 p

= variable used for the pepper model, w = variable used for the watermelon model, = variable used for both crops. a Planting and harvesting dates for drip and Days after transplant (DAT) represent middle of each 14 day water balance period. b Planting and harvesting dates for drip and Top 10 cm soil moisture in the rowmiddle. c Planting and harvesting dates for drip and number of days with rainfall >1 mm. w,p

(lysimeter-based Kc ) and the estimated values (SVM-Kc ). The loss function can be described by introducing (non-negative) slack variables  i ,  i * (Vapnik, 1995) to measure the deviation of training samples outside the ␧-insensitive zone (Fig. 1). Such formulation imposes sparseness in the solution as the errors less than ε are

∗

i i ≥ 0 where, f (x) = w, x + b, w, x denotes the dot product of w and x, x is the input vector, w is the weights vector norm, ε is Vapnik’s insensitive loss function (Vapnik, 1995), C is a cost parameter, and b is a bias. Cost parameter determines the trade-off between the complexity of function and the value up to which the deviation larger than ε are tolerated. The first term of the objective function (Eq. (1)) is a regularization term which avoids the ill-posedness of the estimation problem (Tychonoff and Arsenin, 1977; Gill et al., 2006). The second term is a ␧-insensitive loss function (Vapnik, 1995; Chu et al., 2001), which represents the discrepancy between the actual measurement

Fig. 1. Slack variable ( i ) and ε-insensitive loss function for the SVM model.

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ignored (Vapnik, 1995; Chu et al., 2001). This procedure, thus, has computational advantages. The optimization problem in Eq. (1) is solved in its dual form using Lagrange multipliers (˛ and ˛* ). Writing Eq. (1) in dual form and differentiating with respect to primal variables (w, b, , *) (Vapnik, 1995) gives: Maximize L 

w(˛∗ , ˛) = −ε

(˛i + ˛∗i ) +

i=1

yi (˛i − ˛∗i )

i=1

  1  (˛i − ˛∗i )(˛j − ˛∗j ) xi , xj , 2 L

L

−

L 

(2)

i=1 j=1

Subject to, L 

(˛∗i − ˛i ) = 0,

˛i , ˛∗i ∈ [0, C],

i=1

The approximating function is then written as: f (x) =

N 





(˛∗i − ˛i ) x, xi + b,

(3)

i=1

where, xi ’s are support vectors and ., . indicates dot product. The sample points with non-vanishing coefficients fit the data, which are called support vectors. Support vectors carry all the information necessary to determine the optimal solution. Non-linearity is introduced by preprocessing the training data into a high dimension feature space through the kernel function. Eq. (3) is then upgraded to, f (x) =

N 

(˛∗i − ˛i )k(x, xi ) + b,

(4)

i=1

where, k(x, xi ) is the kernel function that approximates the transformation of input data into high dimension feature space and corresponding dot product in the feature space. Selection of SVM parameter is crucial step in the SVM modeling. The “kernlab” package (Karatzoglou et al., 2004) included in the “R” software (R Core Team, 2012) was used for the SVM model in this study. The Radial basis function (RBF) kernel, known to give better performance than other kernels (Dibike et al., 2001; Scholkopf and Smola, 2002; Asefa et al., 2006; Khalil et al., 2006), was used in this study. The SVM model parameters used here were C and kernel parameter (sigma) (Kuhn, 2008). They were estimated from the 10-fold cross validation technique by using a wide range of model parameters (feasible parameter space) in the training phase using the “caret” package (Kuhn, 2012) in “R” software (R Core Team, 2012). The parameters optimization was confirmed when the parameters for the best model lied within the defined range. Because of this, the variation in SVM parameters is not likely to affect our model results. Further details of the SVM model can be found in Vapnik (1995). The performance of the SVM model was evaluated by comparing the SVM results with the ANN and the RVM. 2.4. Artificial neural network (ANN) The ANN is an information processing system, consists of interconnected group of artificial neurons, and works in the same way as biological neurons and is capable of understanding the complex nonlinear relationship between the response (e.g. Kc ) and predictors (e.g. hydro-climatic data) (Khalil et al., 2005). The ANN structure consists of three distinctive layers called input, hidden, and output. The input layer introduces data (e.g. hydro-climatic

variables) to the ANN model, the hidden layer processes the data, and the results are produced in the output layer (Patterson, 1996; Gurney, 1999). The output signal at each neuron is computed from nonlinear transformation function to the sum of the weighted inputs before it passes to another layer. A sum of the weighted input is computed from the dot product between the input vector, x = [x1 x2 . . .. . .xn ] and corresponding weight vector, w = [w1 w2 . . ..wn ], and the output is evaluated as an activation function of the weighted input. The optimal values of the weights are determined by the minimization of the objective function that measures the difference between the actual and predicted values (Rumelhart et al., 1986). Further details on ANN can be found in Schalkoff (1997) and Govindaraju and Rao (2000). The “nnet” package (Venables and Ripley, 2002) included in the “R” software (R Core Team, 2012) was used in this study for developing the ANN model, which is a feed-forward neural network with a single hidden layer. The ANN model parameters in this package are decay and size (number of units in hidden layer) (Kuhn, 2008), which were optimized using the 10-fold cross validation technique for a wide range of values using the “caret” package (Kuhn, 2012) in “R”. 2.5. Relevance vector machine (RVM) For the given input-target pair {xi , ti }N in the training data set, i=1 the RVM model learns the dependency of targets on the inputs such as hydrologic parameters, time, and climate parameters with the objective of making accurate predictions of t (Kc in this study) for the previously unseen values of x (Tipping, 2000, 2001). Target ti is a sample from the model (yi ) with the additive noise (εi ), which has a mean value of zero with variance  2 (Tipping, 2001). ti = y(xi ; w) + εi

(5)

The unknown function y is the product of design matrix (˚) and weight parameter (w). In the vector form, Eq. (5) can be expressed as, t = ˚w + ε.

(6)

An independent Gaussian noise is assumed (Tipping, 2001).  Thus, p(ti x) ∼N(y(xi ; w),  2 ) and the likelihood of complete dataset is written as, p( t| w,  2 ) = (2 2 )

−N/2

exp{−

 1  t − ˚w2 }. 2 2

(7)

The maximum likelihood estimate of w and  2 in Eq. (7)may suffer from overfitting (Tipping, 2001). To avoid this, w is constrained with mean zero Gaussian prior probability, which results in majority of w being zero. This constrain makes the RVM model sparser than the SVM model (Tipping, 2001). The posterior covariance and mean of w, estimated from Bayes’ rule (Tipping, 2001) are ˙ = −1

and  =  −2 ˙˚T t, respectively, and A = diag(˛0 , ( −2 ˚T ˚ + A) ˛1 , ........, ˛N ), where ␣ is uniform hyperpriors and diag(. . .) is a diagonal matrix. The ˛ and  2 are estimated from an iterative reestimation formula (Tipping, 2001) given by, ˛new i

=

i 2i

2 new

and ( )

  t − ˚2 =  , N−

(8)

i

i

where  i = 1 − ˛i ˙ ii . The term i is the ith posterior mean weight and N is the number of data examples (length of data set). The ˙ ii is ith diagonal element of the posterior weight covariance computed with the current ˛ and  2 . The learning algorithm proceeds by iterative process of Eq. (8) together with updating the posterior

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Table 3 Mean absolute error (MAE), mean square error (MSE), and coefficient of determination (r2 ) for the SVM model for each input set for training and testing phases of pepper and watermelon. The input sets are described in Table 2. Input set

Watermelon

Pepper Training

Input 1 Input 2 Input 3 Input 4 Input 5 Input 6 Input 7 Input 8 Input 9

Testing 2

Training



Testing

MAE

MSE

r

MAE

MSE

r

MAE

MSE

r

MAE

MSE

r2

0.005 0.005 0.008 0.020 0.012 0.018 0.005 0.008

0.036 0.036 0.035 0.033 0.033 0.036 0.041 0.036

0.48 0.49 0.49 0.53 0.53 0.49 0.41 0.49

0.028 0.029 0.037 0.034 0.033 0.031 0.026 0.030

0.034 0.035 0.036 0.041 0.039 0.037 0.034 0.037

0.71 0.68 0.69 0.63 0.69 0.68 0.71 0.68

0.029 0.031 0.017 0.014 0.011 0.015 0.049 0.015 0.017

0.095 0.044 0.052 0.050 0.048 0.550 0.058 0.059 0.059

0.34 0.80 0.72 0.74 0.70 0.69 0.59 0.48 0.53

0.256 0.335 0.388 0.362 0.237 0.224 0.427 0.485 0.213

0.205 0.210 0.206 0.203 0.144 0.137 0.225 0.253 0.116

0.65 0.69 0.82 0.82 0.82 0.80 0.81 0.80 0.82

statistics  and , until suitable convergence criteria is satisfied (Tipping, 2001). The predictions for new input (x* ) are made based on the posterior distribution over the weights, conditioned on the maximizing values ˛ and  2 .



p(t∗ t, ˛,  2 ) = N(t∗ y∗ , ∗2 ) , where y* = T (x* ), ∗2 =  2 + (x∗ )T ˙ (x∗ ), and is a basis function. Further details of the RVM model can be found in Tipping (2000, 2001). The “kernlab” package (Karatzoglou et al., 2004) included in the “R” software was used for the RVM model, which has only one model parameter (sigma) (Kuhn, 2008). The parameter for the RVM model was optimized using the 10-fold cross validation technique for a wide range of parameter values using the “caret” package in “R”. 2.6. Model formulation A wide range of input variables consisting of hydrological, meteorological, soil, and time were used to develop the model. Selection of input variables was based on the current understanding of the physical processes affecting Kc (Allen et al., 1998; Li et al., 2003; Nagler et al., 2005; Rijal et al., 2012). Data from the drip and sub-irrigation systems were pooled, and different combinations of input sets were developed for each crop: pepper and watermelon (Table 2). Kc for each bi-weekly (14 days) period was computed from the average Kc values from four lysimeters for the drip system and two lysimeters for the sub-irrigation system. Data from a total of seven pepper growing seasons and three watermelon seasons were used for this study. For each crop, the data were divided into two parts, training and testing. The model was developed during the training phase and the predictive ability of the trained model was evaluated in the test phase. Dividing data into two sets, training and testing is widely practiced in machine learning regression including the SVM modeling (Asefa et al., 2006; Kalra and Ahmad, 2009). The i.i.d. assumption did not affect the division of the data since the subset of i.i.d. data retained i.i.d. properties. The data were divided such that each phase had the data for both irrigation systems (wetting types) and growing seasons (spring and fall). This reduces bias and retains similar distribution of predictors in the training and test phases. For pepper, a total of 72 data points consisting of both drip (spring 2006 and fall 2003, 2004, 2006, and 2008) and sub-irrigated (spring 2006 and 2007, and fall 2003, 2004, 2006, and 2007) lysimeters were used for the training phase, and a total of 19 data points consisting of drip lysimeters for spring and fall 2007, and the sub-irrigated lysimeters for fall 2008 were used for testing. For watermelon, 24 data points comprising both drip and sub-irrigation system for spring 2004 and

2

2

2005 were used for training, and the eight data points consisting of both irrigation systems for spring 2003 were used for testing. The SVM model was developed for each input set (Table 2) and the best set was selected based on the coefficient of determination (r2 ), mean square error (MSE), and mean absolute error (MAE) for Kc values in the test phase. The same data and goodness of fit were used for the ANN and RVM models. Both MSE and MAE are the measure of how close the prediction is from the actual values while r2 measures how well the model fits the data as proportion of total variation (Cameron and Windmeijer, 1997). MAE is the most natural measure of average error magnitude, while MSE is a widely used statistic in environmental literature (Willmott and Matsuura, 2005). 3. Results and discussion 3.1. SVM predictions For pepper, an erect crop, the model performance statistics for the test phase for all input sets (Table 3) for the SVM model show that input set 1 (r2 = 0.71, MAE = 0.028, and MSE = 0.034) and 7 (r2 = 0.71, MAE = 0.026, and MSE = 0.034) produced better results than other sets. The input set 7 was selected because this was the most parsimonious model, requiring only three readily available variables (DAT, water table depth, and rainfall events) as opposed to 15 required for input set 1 (Table 2). For watermelon, a vine crop, similar r2 were obtained from the sets 3, 4, 5, and 9 (r2 = 0.82) (Table 3). Set 9 had the smallest MAE and MSE (MAE = 0.213, MSE = 0.116) and fewest number of variables; therefore, this set was selected. The best input set (set 9, Tables 2 and 3) for watermelon had the same three variables as used for pepper: DAT, water table depth, and number of rainfall events. Similar to field conditions, lysimeters were drained after a significant rainfall event, bringing the water table depth close to the pre-rainfall condition. The drainage combined with the fact that the Kc was estimated on a bi-weekly basis and not daily or smaller time step reduced the dependency of the water table depth on rainfall. The model developed here is robust since the best set had the same input variables for both crops. These variables are consistent with the current understanding of the physical processes affecting ETc and Kc (Allen et al., 1998; Rijal et al., 2012). The SVM model was observed to be more sensitive to parameter C compared to sigma. Since we optimized the model parameters from the 10-fold cross validation technique over a wide range of parameter values, the variations in SVM parameters did not affect the results. Both best sets had variables that affect transpiration as well as evaporation and, therefore, ETc and Kc . Although soil moisture affects the rate of transpiration, it is only when the soil moisture is below a threshold value that causes stomata closure reducing

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0.0

-0.2

0 (a)

20

40 DAT 60

80

100

-0.4 -0.6 -0.8 -1.0

Drip-irrigation Sub-irrigation

-1.2

Water table depth (m)

Water table depth (m)

0.0

-0.2

0 (b)

20

40 DAT 60

80

100

-0.4 -0.6

Drip-irrigation Sub-irrigation

-0.8 -1.0 -1.2

Fig. 2. Average water table depths for the drip and sub-irrigated lysimeters for (a) pepper and (b) watermelon. Days after transplant (DAT) represent the middle of each 14-day water balance period.

the plant transpiration. The soil moisture in the plastic mulched bed within the lysimeter was at optimum and therefore, it was not a significant variable to explain variability in transpiration and hence Kc . Transpiration increases with the plant growth which is captured by DAT. The leaf area index (LAI) had strong correlation with DAT for pepper for spring and fall 2007 (Pearson correlation coefficient = 0.67; p < 0.001), the only two seasons when LAI was measured. Although LAI was not included in this model, DAT was found to be a reasonable surrogate. Furthermore, requiring LAI as a predictor variable also restricts the use of this model as it is not commonly measured and readily available. The water table depth is an important factor for controlling soil moisture (Kalita and Kanwar, 1992) and, therefore, evaporation from the row-middle areas (Shukla et al., 2012, 2014b; Shrestha and Shukla, 2014). The higher Kc for the sub-irrigated system compared to drip was mainly due to difference in water table depths. The use of water table depth implicitly generalizes the model for the irrigation method as the water table depth for sub-irrigation was statistically higher than that for the drip system for both pepper (Fig. 2a; p < 0.001) and watermelon (Fig. 2b; p < 0.001). The drip system irrigates only the mulched bed and, therefore, does not contribute significantly to evaporation. Evaporation in drip irrigation is further limited due to a deeper water table that limits the capillarybased upflux in the row-middles (Jaber et al., 2006). Sub-irrigation saturates the V-ditch, and the water applied moves rapidly to the water table because of high saturated hydraulic conductivity of the soil (Ksat = 15–50 cm/h; USDA, 1998). For sub-irrigation, the water table is maintained at approximately 0.5–0.6 m from the ground surface, which results in near saturation of the row-middle soil surface due to capillary fringe reaching the surface (Jaber et al., 2006). The effect of water table depth was also reflected in the model development phase. Results showed the average r2 in the test phase decreased by 21% when the water table depth was excluded from the input data sets for pepper. A rainfall event is another important variable that affects the surface soil moisture and evaporation. Rainfall wetting, random in nature, wets the field differently than irrigation. Rainfall wets the open soil surface and brings the soil moisture to near saturation, resulting in evaporation from the row-middle area at the potential rate (Wilson et al., 1997). During rainfall, a fraction of the water is also intercepted by plants and the mulched beds, which can evaporate directly. The interception varies depending on the plant canopy and other factors and has been shown to vary from 9% to 30% (Lloyd et al., 1988; Dunkerley, 2000). Unlike rainfall, sub-irrigation wets the soil from the bottom and does not directly wet the surface as the water is applied to the shallow ditches. Both irrigation methods in this study do not wet the plastic mulch and plant canopy from where it can directly evaporate. In addition, a heavy rainfall can temporarily cause a disproportionate increase in the water table (Jaber et al., 2006), resulting in near saturation of the soil surface. This effect is more pronounced for sub-irrigation compared to drip because of shallow water table for the former. Since the

distribution of rainfall events between spring and fall growing seasons is distinctly different, use of number of rainfall events implicitly generalizes the model for the growing seasons and captures the effect of surface soil moisture on evaporation. The crop cover for pepper is mostly limited to the mulched bed; therefore, evaporation from the row-middle areas constitutes considerable fraction of ETc throughout the growing season, especially for the sub-irrigated systems. However, the effect of rainfall wetting on soil evaporation for watermelon is dominant when the crop cover is low (before the end of development stage) as it almost shades the row-middles after the development stage, suppressing soil evaporation. The number of rainfall events is also an indicator of solar radiation, temperature, and humidity because cloudy conditions before, during, and after rainfall result in decreases in radiation and temperature and increases in humidity. These climatic variables have been shown to be important variables affecting Kc (Allen et al., 1998; Li et al., 2003; Shukla et al., 2014a, 2014b). The selection of input variables for the development of the SVM model in this study is consistent with literature (Doorenboss and Pruitt, 1977; Nagler et al., 2005; Zhou and Zhou, 2009; Rijal et al., 2012), which shows the importance of meteorological, hydrologic, agronomic, and other factors in influencing ETc as well as Kc . Fig. 3 shows the SVM-predicted Kc versus actual values in the training and test phases for pepper. In the training phase, most of the data points scatter about 45◦ line, indicating good model performance. In the test phase, the model overestimated for small Kc (<0.6) and underestimated for high Kc values (>1.3). The range 0.6–1.3, which covers most of season, was accurately predicted. Small Kc values were associated with the initial crop stage, when soil evaporation was dominant. The Kc for the initial stage highly depends on the number of rainfall events (Hunsaker, 1999; Benli et al., 2006; Araya et al., 2011; Shukla et al., 2014a). The first month of the fall growing season (September–October) occurs during the wet period of Florida, while the spring growing season occurs entirely within the dry period. This introduces high variability in the initial Kc values which resulted in high model uncertainty. Underestimation of higher Kc values in the test phase (Fig. 3b) could be due to small numbers of unusually high Kc values in the dataset which resulted in insufficient data to train the model and, thus, relatively larger prediction errors for the higher Kc values in the test phase. These unusually high Kc values are rare occurrences so it is not likely to affect the model predictions and ETc estimates. Most of the SVM predictions for both the training and test phase are populated about 45◦ bisectors, demonstrating good model performance. Fig. 4 shows predicted Kc versus actual values for the training and test phases for watermelon. All points were situated near a 45◦ line with the slope of regression line being close to the slope of the bisector. Pepper being grown in two seasons (spring and fall) captured more of the expected variability compared to watermelon that was grown only during the spring season. Therefore, the SVM model for pepper is more robust as it covers both seasons that have

N.K. Shrestha, S. Shukla / Agricultural and Forest Meteorology 200 (2015) 172–184

1.5

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(a)

(b)

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Predicted Kc

1.0

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0.0

0.0 0.0

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1.0

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Actual Kc

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Actual Kc

Fig. 3. The SVM-predicted Kc versus actual (lysimeter) values for pepper for (a) training phase and (b) test phase. The solid line is a regression line, and dotted line is a 45◦ bisector.

1.5

(a)

1.0

(b)

1.0

Predicted Kc

Predicted Kc

1.5

0.5

0.5

0.0 0.0

0.5

1.0

1.5

0.0 0.0

0.5

1.0

1.5

Actual Kc

Actual Kc

Fig. 4. The SVM-predicted Kc versus actual (lysimeter) values for watermelon for (a) training phase and (b) test phase. The solid line is a regression line, and dotted line is a 45◦ bisector.

Wind speed- spring Solar radiation - spring

100 95 90 85 80 75 70 65

Temperature ( °C)

30 25 20 15 10 0

20

40

DAT

60

80

100

Fig. 5. Seasonal average bi-weekly (14 days) temperature and relative humidity (RH) for spring and fall seasons. Days after transplant (DAT) represent middle of each 14-day water balance period.

Wind speed (m/sec)

Temperature-fall RH- fall

Relative humidity (%)

Temperature-spring RH-spring

were different with the spring being higher (Fig. 6). Distribution of rainfall events are distinctly different for the two seasons, especially during the first two bi-weekly periods (DAT 0–27) of the fall season, which occur within the wet period (June–October) in Florida (Fig. 7). Despite this diversity in climatic conditions between two seasons, the SVM model performed well.

Wind speed- fall Solar radiation- fall

6

300

5

250

4

200

3

150

2

100

1

50

0

0 0

20

40

DAT

60

80

Solar radiation (w/m2)

significantly diverse climatic conditions and, therefore, evaporative demand. Spring in south Florida starts with cooler temperatures and ends with warm conditions. The fall season starts with warm conditions similar to the end of spring season, and ends with the cooler conditions similar to the start of spring (Fig. 5). The relative humidity (RH) for spring is consistently lower than that in the fall (Fig. 5). Solar radiation and average wind speed for the two seasons

100

Fig. 6. Seasonal average bi-weekly wind speed and solar radiation for spring and fall seasons. Days after transplant (DAT) represent middle of each 14-day water balance period.

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In the test phase (Fig. 4b), a large discrepancy between the predicted and the actual values for watermelon was observed for small Kc values, but it was close to the actual values for the mid and high Kc values. The large error for small Kc was due to high uncertainty in Kc during the initial growth stage when evaporation is a dominant part of ETc . This high uncertainty was associated with the complex interaction of rainfall, soil moisture, humidity, and temperature. Figs. 3 and 4 show that the SVM model accurately predicted Kc from the same input variables for both crops with large differences in canopy cover. Since the model works well for both crops with small (pepper) and large (watermelon) canopy cover, it is likely that the SVM model will predict Kc for most crops as a function of crop growth (DAT), hydrologic (water table depth), and climate (rainfall events) factors. This implies that the developed model is robust and parsimonious and should be applicable to other crops for different growing environments for improved prediction of Kc . To predict Kc using the SVM model, DAT, water table depth, and rainfall events are needed. To use the SVM model, the package “kernlab” and “caret” should be installed and loaded in “R”. Using the calibrated (trained) SVM model, Kc for the new input sets can be predicted. Upon the availability of data, a similar model can be developed for a specific hydro-climatic conditions.

difference between the SVM-predicted and actual Kc values. On the other hand, FAO-56 Kc values were statistically lower than both actual (p = 0.016) and the SVM-predicted values (p = 0.018); this explains the underestimation of ETc values by the use of FAO-56 Kc values for sub-irrigated pepper. Results for sub-irrigated watermelon in Table 5 show similar values as that of Table 4 for pepper. Like pepper, the actual Kc values were not statistically different (p = 0.393) from the SVM-predicted Kc values indicating better performance of the model with predicted ETc being 18% less than actual. In contrast, the FAO-56 values were statistically lower (p = 0.025) than the actual Kc values and their use resulted in 46% underestimation of ETc for watermelon. High uncertainty in the initial stage Kc , mainly due to soil evaporation caused by random wetting from rainfall when the plant cover was low, resulted in high error in ETc . When the initial stage was not considered, the error in the SVM-based ETc was only 2.6% (SVM = 176.25 mm and actual = 180.99 mm). Compared to this low error from the SVM model, the underestimation of ETc by the use of FAO-56 Kc was 41.7%. Such low level of error indicates usefulness of the SVM model for predicting local Kc . To scale up the magnitude of these errors, we applied the SVM predicted and FAO-56 Kc values to a vegetable farm (112 ha) where pepper (plastic mulch) is a primary crop. The SVM model predicted ETc was closer to actual ETc by 49.3 mm, compared to FAO-56. This improved prediction was 70% of the measured surface flow (70.7 mm) for a farm, an indication that improvement in ETc prediction can improve the predictions of other water balance components (e.g. surface flows) in hydrologic models. In absence of site-specific Kc or ETc data, the commonly used technique is to adjust other model parameters to achieve satisfactory calibration. This transfer of errors leads to erroneous estimates of water balance components such as groundwater storage and flows. The observed performance of the SVM model clearly shows that it can be effectively used to improve ETc estimates and improve hydrologic model predictions at the watershed scale. Use of Kc from the SVM model, if incorporated in the physically based hydrologic models such as MIKE-SHE/MIKE 11 (Vázquez and Feyen, 2003; Jaber and Shukla, 2012), can improve simulations by realistic representation of the water balance components.

3.2. Comparison of the SVM, actual, and literature Kc and ETc

3.3. Comparison with the ANN and RVM models

Table 4 shows actual, SVM, and FAO-56 Kc and ETc values for the sub-irrigated pepper for fall 2008. The seasonal time-weighted average Kc from the SVM model (0.94) is much closer to actual value (1.09) than FAO-56 (0.69). The SVM predicted Kc values provided a better estimate of ETc than the FAO-56 with only 11.2% error compared to an error of 35.3% when the FAO-56 Kc values were used. The errors associated with the SVM and the FAO-56 pepper Kc values accounted for 16% and 50% of the seasonal rainfall, showing the implications of error in ETc predictions for a water budget. Low error from the SVM model highlights its applicability for developing local Kc values in diverse climatological, hydrological, and management conditions. There was no statistical (p = 0.14)

To assess the relative performance of the SVM model, results from the SVM model for both crops were compared with two other machine learning models, ANN and RVM. The ANN and RVM models were developed using the same training and test data set used for the SVM model. Results show that the SVM model consistently outperformed the ANN and the RVM models for both crops during the test phase (Table 6). Fig. 8 shows the pepper Kc predicted from the ANN model for the training and test phase. The performance of the ANN model in the test phase (r2 = 0.66, MAE = 0.041, and MSE = 0.031) was poorer than the SVM model (r2 = 0.71, MAE = 0.026, and MSE = 0.034), but the pattern of prediction was similar. Similar to SVM model (Fig. 3), the ANN model

60

6

Rainfall (mm)

50 40

5 4

30

3

20

2

10

1

0

Rainfall events

Spring rain Fall rain Rain events-spring Rain events- fall

0 7

21

35

49 DAT

63

77

91

Fig. 7. Seasonal average bi-weekly (14 day) rainfall (mm) and rainfall events (number of events during individual bi-weekly periods) for spring and fall seasons. Days after transplant (DAT) represent middle of each 14-day water balance period.

Table 4 Actual (lysimeter), the SVM-predicted, and the FAO-56 Kc and ETc values for sub-irrigated pepper for fall 2008. Stages

Initial Development Mid-season Late Average/total a

Adjusted for plastic mulch.

Kc

ETc (mm) a

Actual

SVM

FAO-56

Actual

SVM

FAO-56

0.78 0.92 1.31 1.33 1.09

0.78 0.86 1.01 1.14 0.94

0.48 0.66 0.84 0.72 0.69

47.0 57.9 70.3 29.6 204.7

46.9 55.1 54.3 25.6 181.8

28.9 42.1 45.3 16.1 132.5

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Table 5 Actual (lysimeter), the SVM-predicted, and FAO-56 Kc and ETc values for sub-irrigated watermelon for spring 2003.

Initial Development Mid-seasonb Average/total a b

ETc (mm)

Kc

Stages

Actual

SVM

FAO-56a

Actual

SVM

FAO-56

0.75 0.85 1.33 0.98

0.20 0.82 1.33 0.78

0.29 0.51 0.73 0.51

50.72 138.89 42.10 231.71

13.58 134.29 41.96 189.83

19.66 82.67 22.91 125.24

Adjusted for plastic mulch. Incomplete stage.

Table 6 Comparison of the SVM model results with the ANN and RVM for pepper and watermelon Kc based on mean absolute error (MAE), mean square error (MSE), and coefficient of determination (r2 ). Model

Watermelon

Pepper Training

SVM ANN RVM

Testing 2

Training 2

Testing

MAE

MSE

r

MAE

MSE

r

MAE

MSE

r

MAE

MSE

r2

0.005 0.000 0.004

0.041 0.038 0.046

0.41 0.45 0.33

0.026 0.041 0.077

0.034 0.031 0.061

0.71 0.66 0.28

0.017 0.000 0.017

0.059 0.002 0.080

0.53 0.98 0.31

0.213 0.333 0.227

0.116 0.394 0.078

0.82 0.43 0.70

(Fig. 8) overestimated the lower Kc values and underestimated the high Kc values in the test phase. The RVM-predicted pepper Kc values considerably deviated from the bisector (Fig. 9), indicating poor performance for both the training and test phases (Table 6). The deviation from the bisector was more pronounced in the test phase (r2 = 0.28, MSE = 0.061, and MAE = 0.077). The last four Kc values

in the test phase were underestimated, while all remaining data points were highly overestimated. For watermelon, the ANN model performed well in the training phase but, in contrast, the predictions for the test phase highly deviated from the bisector (Fig. 10), indicating poor performance (r2 = 0.43, MSE = 0.394, and MAE = 0.333) (Table 6). Compared to the

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2

(b)

1.0

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0.0

1.5

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1.5

Fig. 8. The ANN-predicted pepper Kc versus actual (lysimeter) values for (a) training phase and (b) test phase. The solid line is the regression line, and dotted line is a 45◦ bisector.

1.5

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(a)

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Predicted Kc

1.0

(b)

1.0

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1.0 Actual Kc

1.5

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1.0

1.5

Fig. 9. The RVM-predicted pepper Kc versus actual (lysimeter) values for (a) training phase and (b) test phase. The solid line is the regression line, and dotted line is a 45◦ bisector.

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1.5

Fig. 10. The ANN-predicted watermelon Kc versus actual (lysimeter) values for (a) training phase and (b) test phase. The solid line is a regression line, and dotted line is a 45◦ bisector.

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Predicted Kc

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Fig. 11. The RVM-predicted watermelon Kc versus actual (lysimeter) values for (a) training phase and (b) test phase. The solid line is a regression line, and dotted line is a 45◦ bisector.

SVM model, the RVM model resulted in poor predictions for both the training (r2 = 0.31, MSE = 0.08, and MAE = 0.017) and test phases (r2 = 0.70, MSE = 0.078, and MAE = 0.227) (Table 6 and Fig. 11). Given the complexity and variability in ET, the ANN model may converge to local optima rather than global, resulting in poor prediction compared to the SVM model which ensures global optimization. The RVM model is sparser than the SVM model and may use fewer data points from the training set compared to the SVM model. The fewer data points in the RVM model may not be enough to capture the large variability in hydrologic and climatic conditions observed in this study resulting in less accurate prediction compared to the SVM model. Overall, the SVM model showed superior performance over the ANN and RVM models for both crops. Using the model-predicted Kc , ETc values were computed for the SVM, ANN, and the RVM models for both crops. The r2 between the observed and predicted ETc for pepper were 0.61, 0.62, and 0.51 in the training phase and 0.48, 0.42, and 0.54 for the test phase for the SVM, ANN, and RVM models, respectively. The respective values of MAE for pepper were 0.42, 0.76, and 0.73 mm in the training phase and 0.76, 0.69, and 6.64 mm in the test phase. For watermelon, the r2 values were 0.72, 0.99, and 0.59 for the training phase and 0.75, 0.29, and 0.66 for the test phase for the SVM, ANN, and RVM models, respectively. The respective values of MAE for watermelon were 0.08, 0.04, and 0.35 in the training phase and 15.22, 20.83, and 16.63 mm for the test phase. Relatively better performance was obtained from the SVM model compared to ANN and RVM models. Although the SVM-based ETc estimates are relatively less accurate compared to SVM-based Kc estimates, they are

nonetheless, still significantly better than FAO-56 estimates. The superior performance of the SVM model indicates that the use of the SVM model with readily available hydro-climatic data can improve the ETc estimates.

4. Summary and conclusions The support vector machine, a data-driven model based on statistical learning theory, was successful in developing a generic model for predicting Kc and ETc for two crop types for different irrigation and climatic conditions. The model was developed from five years of lysimeter data for an erect crop (pepper) and three years of data for a vine crop (watermelon) with the drip and sub-irrigation grown on plastic mulched beds. The SVM model was trained and tested on the data set consisting of two irrigation systems and two distinctly different seasons, spring and fall. The SVM model performed well for both pepper (r2 = 0.71, MSE = 0.03, MAE = 0.03) and watermelon (r2 = 0.82, MSE = 0.11, MAE = 0.21) in predicting Kc as a function of days after transplant (DAT), water table depth, and number of rainfall events. The model identified key variables affecting soil evaporation and plant transpiration in a relatively shallow water table environment. These variables are consistent with the current understanding of the physical processes affecting ETc and Kc . Day after transplant represents the plant growth and, therefore, transpiration. The rainfall events and water table depth captured the effect of surface soil moisture in the row-middles on evaporation.

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The SVM model was robust as it accurately predicts Kc for two distinctly different crop types (vine and erect) with regard to crop cover using the same predictors. For pepper, the model performed well for two distinctly different seasons with the diverse rainfall, humidity, and wind speed conditions. For both crops, the SVMpredicted Kc values were not statistically different from the actual values derived from lysimeters. Use of SVM-predicted Kc estimated pepper ETc with only 11.2% error compared to 35.3% for the commonly used FAO-56 Kc values. For watermelon, when the initial stage was excluded, the SVM model predicted ETc with 2.6% error compared to FAO-56 for which the error was 41.7%. Improved accuracy was demonstrated when the SVM and FAO-56 Kc values were used for farm-scale ETc predictions for pepper grown with plastic mulch. The use of the SVM model improved ETc estimates by 49.3 mm (compared to FAO-56) and this improvement was 70% of the farm-scale surface flows. This improvement in ETc from the SVM model will improve the accuracy of other water balance components (surface and sub-surface storages and flows) in hydrologic models. The uniqueness of this study is that it can predict farm-scale Kc (ETc ) accurately for both vine and erect crops grown on two irrigation systems and climatic conditions using readily available hydro-climatic data used by farmers in managing their farms. The performance of the SVM model was superior to the ANN and RVM models. Since the SVM model worked well for both erect and vine crops, it is expected to work well for other crops. The SVM model developed in this study will be useful to accurately quantify the crop water use and develop effective irrigation plans for the conservation of water resources at a farm as well as on a regional scale. Accurate simulation of ETc is an important aspect of hydrologic modeling, because it is the largest component of the watershed-scale cycle for most regions of the world. Reduction of error in predicting ETc in hydrologic models will reduce uncertainty in streamflow and groundwater recharge, two key variables targeted in hydrologic modeling for a wide variety of applications. Similar SVM-based models need to be explored for other crops with the different surface (e.g. open field), hydrologic, climatic (e.g. arid), and management conditions. References Agam, N., Evett, S.R., Tolk, J.A., Kustas, W.P., Colaizzi, P.D., Alfieri, J.G., McKee, L.G., Copeland, K.S., Howell, T.A., Chavez, J.L., 2012. Evaporative loss from irrigated interrows in a highly advective semi-arid agricultural area. Adv. Water Resour. 50, 20–30. Akintoye, H.A., Kintomo, A.A., Adekunle, A.A., 2009. Yield and fruit quality of watermelon in response to plant population. Int. J. Veg. Sci. 15 (4), 369–380. Allen, R.G., Pereira, L.S., Howell, T.A., Jensen, M.E., 2011. Evapotranspiration information reporting: I. Factors governing measurements accuracy. Agric. Water Manag. 98 (6), 899–920. Allen, R.G., Pereira, L.S., Raes, D., Smith, M., 1998. Crop Evapotranspiration. United Nations FAO, Rome, Italy (FAO Irrigation and Drainage Paper No. 56). Allen, R.G., Pereira, L.S., Smith, M., Raes, D., Wright, J., 2005. FAO-56 dual crop coefficient method for estimating evaporation from soil and application extensions. J. Irrig. Drain. Eng. 131 (1), 2–13. Allen, R.G., Pereira, L.S., 2009. Estimating crop coefficients from fraction of ground cover and height. Irrig. Sci. 28 (1), 17–34. Amayreh, J., Al-Abed, N., 2005. Developing crop coefficients for field-grown tomato (Lycopersicon esculentum Mill.) under drip irrigation with black plastic mulch. Agric. Water Manag. 73 (3), 247–254. Araya, A., Solomon, H., Mitiku, H., Sisay, F., Tadesse, D., 2011. Determination of local barley (Hordeum vulgare) crop coefficient and comparative assessment of water productivity for crops grown under the present pond water in Tigray, Northern Ethiopia. Afr. J. Online. Asefa, T., Kemblowski, M., McKee, M., Khalil, A., 2006. Multi-time scale stream flow predictions: the support vector machines approach. J. Hydrol. 318 (1–4), 7–16. Benli, B., Kodal, S., Ilbeyi, A., Ustun, H., 2006. Determination of evapotranspiration and basal crop coefficient of alfalfa with a weighing lysimeter. Agric. Water Manag. 81 (3), 358–370. Beven, K., 1989. Changing idea in hydrology—the case of physically-based models. J. Hydrol. 105 (1–2), 157–172. Bryla, D.R., Trout, T.J., Ayars, J.E., 2010. Weighing lysimeters for developing crop coefficients and efficient irrigation practices for vegetable crops. HortScience 45 (11), 1597–1604.

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