Estimating Potential Evapotranspiration in Maranhão State Using Artificial Neural Networks

The use of technology and planning in agricultural production is essential in Northeastern Brazil, which is the region of the country that most suffers from water shortage. For the best irrigation management, it is necessary to know the potential evapotranspiration rate for water control in order to increase productivity. There are several direct and indirect methods for estimating evapotranspiration, but the standard method recommended by the United Nations Agriculture Organization (FAO) is the Penman-Monteith ( PETpm ) method because it has higher accuracy than other methods. However, it is a difficult method to be used due to the need for a large number of meteorological elements. In this con-text, the objective of this study was to estimate potential evapotranspiration by the Penman-Monteith method in the micro-region of Baixo Parnaíba in Maranhão state using artificial neural networks. Agro-meteorological data were collected daily over 34 years, from 1984 to 2017, and these data were obtained from the NASA/POWER website. Subse-quently, liquid radiation and potential evapotranspiration were calculated by the Penman-Monteith standard method (1998). To predict potential daily evapotranspiration, the Multi-Layer Perceptron (MLP) was chosen, which is a traditional Artificial Neural Network. The period that presented a higher evapotranspiration index was the same one that showed precipitation with a lower volume and higher temperatures. The artificial neural network model that best adapted to estimate PETpm was MLP 2-5-1. It is concluded that artificial neural networks estimate with accuracy and preci-sion the Penman-Monteith daily potential evapotranspiration of the Lower Parnaiba in Maranhão, and potential evapotranspiration can be estimated by the Penman-Monteith method using neural networks with inputs of air temperatures. padrão Penman-Moteith (1998). Para prever a evapotranspiração potencial diária foi escolhida a Perceptron Multi-Layer (MLP), que é uma Rede Neural Artificial tradicional. O período que apresentou um maior índice de evapotranspiração foi o mesmo que apontou precipitações com um menor volume e temperaturas elevadas. O modelo de rede neural artificial que melhor se adequou para estimativa da ETPPM foi a MLP 2-5-1. Com isto, conclui-se que as redes neurais artificiais podem ser utilizadas para a estimativa da ETPPM diária do Baixo Parnaíba maranhense. É possível estimar a evapotranspiração potencial por Penman-Monteith usando redes neurais com inputs de temperaturas do ar.


Introduction
Water shortage in the Northeast region of Brazil is a great problem compared to other regions of the country, and for the expansion of the agricultural sector and the rational use of water it is essential to apply technology and planning for the most efficient irrigation to occur in this region (Andrade Junior et al., 2018;Freitas et al., 2018). Furthermore, increased temperature and reduced precipitation causes increased water demand for crops (Gava et al., 2015) and damages agricultural production.
Establishing the water requirements of a crop in its different stages of development is of fundamental importance for irrigation management to acheive good productivity and rational water use (Costa et al., 2018). For this to occur it is necessary to know the evapotranspiration rate of the plant in order to irrigate with the correct amount, with the goal of increasing production without waste of water.
Evapotranspiration is the process in which water from the earth's surface evaporates into the atmosphere together with the transpiration of plants, being an important part of the hydrological cycle (Alves et al., 2017). The process of evapotranspiration occurs not only from surface water, but also involves the loss of soil water and moist vegetation in the form of water vapor and the loss of water from animals (perspiration) and plants through transpiration. Thus, the study of daily evapotranspiration rates is very useful for establishing water requirements in agriculture (Martins and Rosa, 2019).
To estimate the amount of water required by a plant it is necessary to calculate the potential evapotranspiration, and this can be done by several empirical methods (Andrade Junior et al., 2018). The Food and Agriculture Organization (FAO) recommends Penman-Monteith as a standard method, since it is considered a more accurate model (Allen et al., 1998). What makes it difficult to use is the need for various climatic elements, such as relative humidity, solar radiation, air temperature, and wind speed. This method requires a larger number of weather variables and these elements are often not available due to the small number of weather stations available in many regions, and when these are present they may contain insufficient data (Alves et al., 2017). In the literature there are other empirical methods with fewer requirements for climatic variables (Silva et al., 2018, Caporusso andRolim, 2015), such as Hargreaves-Samani, Solar Radiation, Blaney-Criddle and Priestley-Taylor (Allen et al., 1998), but these methods oftentimes have large deviations. One way to accurately estimate PETpm is by using artificial intelligence, such as through artificial neural networks (ANN).
In the last decade, intelligent computational models have been developed as alternative methods to estimate potential evapotranspiration (Yassin et al., 2016), and ANN is one of the techniques that can be used. ANN are mathematical models that simulate the neural structures of the human brain and build computational intelligence through learning and generalization (Di Domenico et al., 2017). This method has a nonlinear structure that captures the most complex characteristics of the data, which is difficult to achieve using other traditional statistical techniques (Conti et al., 2016;Nunes, 2018).
In recent years, ANN applications have been used in the estimation of PET and the results suggest that artificial neural networks are more accurate than conventional methods (Dai et al., 2009;Abdullahi et al., 2017). However, the use of empirical models of potential evapotranspiration that use only air temperature as input data is relevant in regions with a shortage of climatic data, and in the commercial market there are devices that can measure this variable with low cost, such as thermo-hygrometers.
Water shortage and growing demand for food supply increase the importance of developing improved methods for estimating potential evapotranspiration (Patil and Deka, 2016). In this context, the objective of the study was to estimate potential evapotranspiration by the Penman-Monteith method in the micro-region of Baixo Parnaíba in the Maranhão state using artificial neural networks.

Material and Methods
The study area was in the Baixo Parnaíba microregion in the state of Maranhão, composed of 16 localities ( Fig. 1 and Table 1). This region has high potential for farming due to great availability of soils and water resources.
The historical data series includes minimum, maximum, and mean air temperature (°C), global radiation (MJ m 2 day), precipitation (mm), wind speed (m s −1 ), and relative humidity (%) from 34 years (1984 to 2017) of daily measurements. Climatic data were collected from the NASA/POWER website (Stackhouse et al., 2017). This data platform has a spatial resolution of 0.5°latitude-longitude.
where PET -Potential Evapotranspiration, in mm day −1 ; Rn -net radiation, in MJ m −2 day −1 ; G -soil heat flux, in MJ m −2 day −1 ; T -mean daily air temperature, in°C; γ -Psychometric constant (0.063 kPa°C −1 ); Δ -is the humidity pressure curve decline at the air temperature, in kPa°C −1 ; U2 -Average daily wind speed at 2 m, in m s −1 ;  ea -humidity partial pressure, in kPa; es -humidity saturation pressure, daily average, in kPa; Tmax -Maximum temperature (°C); Tmin -Minimum temperature (°C); RH -relative humidity (%); BOC is the shortwave radiation balance (MJ m −2 day −1 ) and BOL is the longwave radiation balance (MJ m −2 day −1 ); Rs is the incident solar radiation (MJ m −2 day −1 ); α is the coefficient of reflection of vegetation; Rso is the incident solar radiation in the absence of clouds (MJ m −2 dia −1 ), and Ra = solar radiation at the top of the atmosphere (MJm −2 dia −1 ). Daily minimum, mean, and maximum air temperature and potential evapotranspiration data were randomly divided into two sets of data values, one for network training (139,000) and the other to evaluate the validation of the trained network (59,613), with a ratio of about 70% of the data for training and 30% for validation, respectively.
The definition of the architecture of networks is the number of neurons per layer and number of layers, and is optimized by the Intelligent Problem Solver tool (IPS) of the Statistica 7 software (Statsoft, 2007). The training itself and the verification of the network and validation of the process were performed by IPS.
To predict daily PETpm, the Perceptron Multi-Layer (MLP) was chosen, which is a traditional Artificial Neural Network that has three stages: input, processing, and output (Fig. 2). Numerous topologies and variations were tested between neurons in the intermediate and outer layers until the most accurate ones were obtained. The ANN input data were the maximum, mean, and minimum air temperatures (°C).
A layered MLP, with H hidden neurons, and an output neuron can be expressed by Eq. (9) in mathematical language.
where Y is the RNA output; Oh is the output value of the Hth occult neuron, with Oh given by (Eq. (10)): where Xn is the RNA output; WNh and Wh are the synaptic weights between hidden neurons and RNA outputs and inputs, respectively, Xo is 1, and Wo and Woh are initial values of RNA training algorithms. The activation function is the logistic sigmoid given by (Eq. (11)): where x is the sample mean that forms the training set.
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Ea = ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Es = ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi

RMSE =
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where Yest i : Variable estimated by RNA; Yobsi: Variable observed; N: Number of data; k: Number of independent variables in the regression. The criterion adopted to interpret performance by the Camargo and Sentelhas confidence index (1997) can be observed in Table 2.

Results and Discussion
The mean values of maximum, minimum and mean air temperatures in the 16 localities in the period 1984-2017 were 32°C, 24°C, and 28°C (Fig. 3) respectively. The mean maximum temperature of the region occassionally reached almost 40°C. In the spatial evaluation, the lowest minimum air temperatures occurred in the Southern region, with values ranging from 22.69°C to 22.99°C. However, the highest maximum air temperature was predominant in Araioses, 30.45°C (Fig. 4).
The highest rainfall occurred in the period from March to April and the lowest in the period from August to November (Fig. 5). In March, the accumulated monthly rainfall was 312 mm, while the month of September had 12 mm. The months that had a higher precipitation rate are considered the rainy season and months with lower rainfall volumes are considered the dry season in the microregion. Passos et al. (2016), in a study of water balance and climatic classification for the Chapadinha-MA region, which is one of the cities of the micro-region observed in the current study, described two well-defined seasons, the rainy season from January to May, with 84% of total annual precipitation in these months, and the dry season from June to December which had 16% of total rainfall.
The increase in air temperature in the Baixo Parnaíba region occurred in August. Between February and July, the mean monthly air temperature was 26°C. In the period from April to October, there was an increase in PET in the micro-region of Baixo Parnaíba (Fig. 6), and the lowest PET of the region occurred in April. October had the highest potential evapotranspiration value of the region, and the period from August to October presented a high PET index which coincided with a lower precipitation volume. Freitas et al. (2018) observed that the months that presented higher precipitation also had the lowest rate of PET, and this was due to the occurrence of rain events wherein air masses create a barrier that hinders the incidence of solar radiation, and as a result evapotranspiration rates are reduced.
The arrangement of the MLP artificial neural network that best adapted to estimate PETpm was MLP 2-5-1, that is, with two input variables, five neurons in the hidden layer and one output. Among the models presented, MLP 2-5-1 was the one that was most able to express in a simplified way all the variation of PETpm. This is because this multi-layered RNA (MLP) method has a more complex (non-linearly separable) solution than methods with only one layer that have the ability to solve only linearlyseparable problems. Tabari and Talaee (2013) indicated that the main difficulty in applying the PM equation is due to the requirement for a wide variety of meteorological data that must be used as input for the calculation of PET. In addition, PET is non-linear, which would make it difficult to accurately affect changes in climatic parameters. Therefore, the use of MLP neural network techniques may have better performance, as these can solve separable nonlinear equations.  In the sensitivity analysis of the input variables of the artificial neural network, the maximum and minimum air temperatures were the highest sensitivities variables in the estimate of PETpm, respectively (Table 3). The maximum air temperature was the one with higher sensitivity, while the minimum air temperature was next in the sensitivity analysis, and the mean temperature did not present sensitivity in the model, probably because it was correlated with these variables.

Statistical indices show that MLP can predict
PETpm with accuracy. Both in training and the MLP test, MAPE was low and R 2 was elevated (Table 4). For example, Pearson's correlation between the actual PETpm data and those estimated by the MLP was 0.955 and 0.959 for training and testing, respectively. The accuracy of the model was good because the value of the Root Mean Squared Error (RMSE) was only 0.331mm and 0.327 mm for training and testing, respectively.
In the training of the network, it was able to explain 91.2% of the variation of the data of the actual PETpm using temperature data as input to the artificial neural network (Fig. 7). The average sensitivity of Penman-Monteith's daily PET estimated by the neural network was 0.8064 mm day −1 . The training sensitivity of 0.8064 mm    In the validation of the PETpm data, the neural network was able to explain 92% of the total variation of the observed data. In general, there was an underestimation of MLP in the estimation of PETpm in all periods (Fig. 8).
The results of the validation (generalization) of the model were similar to those of training. Validation separates the dataset into a training set that is used only to derive parameters that minimize the loss function and a validation set that is used only to evaluate model performance (Jahn, 2018). After each interaction, the training session is interrupted to verify the error.

Conclusions
Artificial neural networks can accurately estimate the daily potential evapotranspiration by the Penman-Monteith method for the Baixo Parnaíba region in Maranhão. This study has shown that it is possible to estimate potential evapotranspiration by the Penman-Monteith method using neural networks with inputs of only air temperatures. Additionally, the artificial neural network underestimated potential evapotranspiration by the Penman-Monteith method, both in training and in testing.