Artificial neural network model for simulation of water distribution in sprinkle irrigation

Modelo em redes neurais artificiais para a simulação da precipitação na irrigação por aspersão R E S U M O A determinação dos coeficientes de uniformidade de distribuição da água dos sistemas de irrigação por aspersão depende, em geral, de ensaios em campo que envolvem tempo e recursos financeiros. Uma alternativa para reduzir custo e tempo é a utilização de simulações. Objetivou-se, com este trabalho, o desenvolvimento de um modelo em redes neurais artificiais (RNA) para simular a precipitação de um aspersor cujos parâmetros de entrada são os valores de pressão de operação, velocidade do vento, direção do vento e diâmetro do bocal do aspersor. Ensaios foram feitos em campo com um aspersor operando em uma malha de 16 x 16 coletores com espaçamento de 1,5 m e diferentes combinações de bocais, pressões e condições de vento. O modelo RNA apresentou bons resultados na simulação da precipitação com correlação de Spearman (rs) entre os dados obtidos em ensaio de campo e os dados simulados, apresentando valores entre 0,92 e 0,97 e índice de concordância de Willmott (d) entre 0,950 e 0,991 para dez ensaios analisados. O modelo RNA mostrou-se promissor na simulação da precipitação em sistemas de irrigação por aspersão.


Introduction
In order to ensure and improve crop yield, the technique of irrigation has been used as a complement or a substitute of natural rainfall, so as to guarantee that the water demand is met.
Among the existing irrigation systems, sprinkle irrigation is one of the most used, since it can adapt to the diversities of soil, topography, crop and the size of the area to be cultivated, which allows a good control of the applied water depth and consequently of its efficiency.
In order to reduce costs and the waste of water and time in the evaluation of sprinkle irrigation systems, simulations can be used to predict the behavior and the results of irrigation.
Many researchers have studied the importance of simulations, such as Faria et al. (2012), who evaluated the applicability of a semiempirical model of Richards & Weatherhead for trials under different wind conditions, and Oliveira et al. (2009), who evaluated the hypothesis of the existence of a linear relationship between the radius of throw of a gun type sprinkler and wind speed.
In the context of the simulations, it is possible to use artificial neural networks (ANNs), which are computational models inspired in the neural structure of intelligent organisms, neurons and synapses, that acquire knowledge through experience.The most important property of ANNs is the ability to learn through examples, by adjusting the weights of the connections between neurons.
ANNs can be applied in many areas, like soil digital mapping based on soil-landscape relationships (Arruda et al., 2013), classification of degradation levels of pastures (Chagas et al., 2009), rainfall erosivity (Moreira et al., 2006), estimation of reference evapotranspiration through air temperature data (Alves Sobrinho et al., 2011), modeling of soil penetration resistance (Santos et al., 2012) or even identification and classification of soybean cultivars by planting region (Galão et al., 2011).
In this context, this study proposed the development of a multilayer perceptron (MLP) neural network model for the simulation of the precipitation of a sprinkler.

Material and Methods
The ANN model was trained and validated using results from field trials at the Ouro Verde Farm, located in the municipality of Brazil (25° 12' 19.26" S;54° 3' 27.93" W;360 m), in order to characterize the spatial distribution of water applied by a sprinkler (Pingo Giro Completo -Fabrimar ® ), operating with different combinations of nozzles, operating pressure, wind speed and wind direction.
The sprinkler was installed at 1.25 m from the soil surface with water nozzle at a vertical distance of 0.60 m from the collectors.In total, 256 pluviometers were installed from the center of the irrigated area (24 x 24 m), at a regular spacing of 1.5 x 1.5 m in both coordinate directions.Figure 1 shows the coordinate system (X,Y) adopted in the trials and the notation of the angle F, which describes the predominant wind direction in relation to the lines of the water collectors used in the field trials.Two kits for water distribution uniformity trials (Fabrimar ® ) were used, which consisted of: 300 pluviometers made of engineering plastic for water collection; 10 graduated cylinders made of transparent thermoplastic, with scale of 0.5 mm, from 0 to 15 mm; 300 bars made of structural aluminum, for the support and fixation of the collectors on the ground.
The operating pressure of each test was controlled using a vertical, glycerin-filled Bourdon tube pressure gauge, with scale of up to 981 kPa, attached to the base of the sprinkler and another one with scale of up to 686 kPa at the beginning of the mainline.The pressure was regulated through a gate valve and the operation time controlled using a digital stopwatch (Casio ® ) Wind speed and wind direction were automatically recorded by a weather station (Davis Vantage Vue), at regular intervals of 5 min, at 2 m from the soil and at a distance of 25 m from the irrigated area.Precipitation measurements were recorded in worksheets, in the form a 16 x 16 matrix, referred to as "A".
Nine combinations of nozzles (2.6 x 2.4; 2.8 x 2.4; 3.0 x 2.6; 3.2 x 2.6; 3.4 x 2.6; 3.6 x 2.8; 3.8 x 2.8; 4.0 x 2.8 and 4.0 x 3.2 mm) were tested at the four pressure values suggested by the manufacturer (196,245,294 and 343 kPa).Since the number of samples necessary for the training of a neural network is unknown (Másson, 1990) and the possibility of success increases as the number of samples increases, three trials were performed for each combination.However, these trials were not three replicates, because wind speed and wind direction changed according to the field conditions during each trial.Thus, 108 trials were conducted, each one with duration of 2 h.
The body of the artificial neuron consisted of a sum function (linear combiner Σ), represented by the weighted sum of the values received by the neuron through the synapses and summed to the bias value that is externally applied and has the effect of increasing or decreasing the output value (v k ) of the body of the neuron (Haykin, 2001).This value generated in the body of the neuron (v k ) passes through an activation function φ(.), which restricts the output amplitude of the neuron and generates an output value (y k ), which represents the net output value of the neuron and will be transmitted to the subsequent neuron, simulating the flow of impulses in the brain.
The neuron can be mathematically described by Eqs. 1 and 2 (Haykin, 2001): A multilayer perception neural network was developed and validated, trained with the backpropagation algorithm, a feedforward function and a sigmoid-type activation function, φ(.), defined by Eq. 3 (Haykin, 2001): In this study, the agreement index (d) was used to compare the values predicted by the model with the observed values.According to Willmott (1981), this index varies from 0 (no agreement) to 1 (perfect agreement) and can be used to measure the accuracy between two models, as shown in Eq. 4. ( ) where a is the parameter defining the slope of the function.
In this study, the dataset of the 108 trials was normalized in the interval from 0 to 1, according to Chagas et al. (2010), and randomly divided into subsets: training subset data (98 trials) and validation subset data (10 trials); the latter was used for the discussion of the results of the model.As in Santos et al. (2012), the strategy of cross-validation was employed in this study, with 20% of the training data used to estimate the performance of the ANN and determine the moment to interrupt the training, in order to avoid excessive training.
For the parametrization, creation, training and validation of the topologies of the neural network in this study, since 31148 topologies of candidate networks were evaluated and the one with the best result was selected, four computational algorithms were developed and executed by the software for neural network simulation SNNS (Zell et al., 1996), in order to test networks with, respectively, one, two, three and four hidden layers.
The ten randomly selected trials for the validation subset were 13,15,41,55,56,85,88,89,93 and 100.Precipitation measurements and field conditions (combination of nozzles, operating pressure, wind speed and wind direction) in which these trials were performed are described below.
Table 1.Tests of adherence to normal distribution for the validation subset of precipitation data ( ) ( ) where: E i -values predicted by the model; O i -observed values; and O -mean of the observed values.

Results and Discussion
The ANN with the best results was composed of one input layer with 4 neurons, one hidden layer with 280 neurons and the output layer with 256 neurons.The fact that the ANN with the best results has only one hidden layer agrees with the results observed by Soares et al. (2014), who estimated soil water retention using ANN, and Zanetti et al. (2008), who estimated reference evapotranspiration using ANNs and concluded that only one hidden layer is sufficient to represent the non-linear relationship between climatic elements and reference evapotranspiration.
According to the results of the analysis and the inferences on data distribution (Table 1), in the normality tests of Kolmogorov-Smirnov and Shapiro-Wilk for the ten trails used in the validation of the ANN model, considering 0.05 significance (α = 0.05), the hypothesis that precipitation data follow normal distribution was rejected.The non-normality of the data is a consequence of the high frequency of values equal to zero, observed in the field trials.
At 0.05 level of significance, the Levene test accepts the hypothesis of homogeneity between observed and simulated data with p-value > 0.05 for seven of the ten trials (Table 2).According to the test of Wilcoxon-Mann-Whitney (a = 0.05 ) (Table 2), used to indicate whether the position measurements of the tested dataset come from similar populations, in eight of the ten trials the observed and simulated data come from similar populations (or the same population), reinforcing the hypothesis that the ANN expressed with good precision the characteristics of the training samples.

Figure 1 .
Figure 1.Scheme of the collection grid and wind direction Figure 2. Sprinkler precipitation for the best correlation coefficient, r s = 0.976, in the Trial 93: observed data (A) and ANN-simulated data (B)