Evaluation and Modelling of Reference Evapotranspiration Using Different Machine Learning Techniques for a Brazilian Tropical Savanna

Abstract

Meteorological elements can affect the environment and cultures differently and may alter the natural development process contributing significantly to climate change. Meteorological variables of the Brazilian Pantanal were studied and used to determine evapotranspiration with fewer variables. It was found that artificial intelligence can substantially improve environmental modeling when alternative prediction techniques are used, resulting in lower project costs and more reliable results. This work tried to find the best combination by comparing machine learning techniques such as artificial neural networks, random forests, and support vector machines. A new model was created that depends on fewer climatic variables compared to the Penman–Monteith method (the standard method for estimating reference evapotranspiration) and can efficiently describe the reference evapotranspiration. Machine learning techniques are highly efficient for modeling environmental systems since they can process large amounts of data and find the best interactions between the parameters involved. In addition, more than 98% accuracy was obtained using fewer variables compared to the standard method when artificial neural networks are utilized.

1. Introduction

There is ample scientific evidence and numerous studies indicating that climate change poses a significant risk to our planet’s natural resources [1,2]. Studies in regions with a rich diversity of animals and plants are of great interest for various analyses [3] because it is possible to examine the frequency and intensity of extreme climate events of short duration, which may result in increased climate oscillations.

Thus, maintaining natural landscapes ensures the regularity of water resource availability [3,4,5]. Some research on evapotranspiration in the Brazilian Pantanal quantifies the volume of water transferred to the atmosphere through soil evaporation and plant transpiration [6].

Researchers have used varied methods, such as direct measurements in micrometeorological towers and estimates based on mathematical models. The results indicate that this hydrological component contributes significantly to the region’s water loss [7,8,9].

It is necessary to carry out continuous and efficient monitoring to understand how the factors affecting climate change are related to the specific biomes of each region [10]. Using reference evapotranspiration (ETo) [11,12], an agrometeorological parameter dependent on various meteorological variables, is essential for the hydrological cycle, since it helps to monitor the water and energy balance continuously.

Accurately estimating reference evapotranspiration (ETo) is essential for many research areas, including climatology and hydrology [13]. ETo measures the atmospheric demand for water, influenced by temperature, relative humidity, solar radiation, and wind speed. In addition, ETo is often used as a benchmark to estimate the actual water demand of crops and ecosystems [14].

In climatology, ETo stands out as an essential indicator of regional and global climate. Studies show climate change affects evapotranspiration, affecting a region’s hydrological cycle and water availability [15]. ETo is often used in climate simulation models to estimate changes in water demand and help predict future climate changes in hydrology. It is used to estimate crops’ and ecosystems’ actual evapotranspiration (ETR), a crucial component of the hydrological cycle. The ETR affects the amount of water available to plants, aquifer recharge, and surface runoff [16]. ETo is often used in hydrological models to estimate the ETR in a given region and help manage water resources more efficiently.

It is important to note that the choice of potential evapotranspiration (ETP) model affects the estimation of ETo and the assessment of aridity [14]. Recent studies have shown that using different ETP models can lead to different values of standardized precipitation and evapotranspiration indices (SPEI) and aridity indices [14,17]. Accurate estimation of ETo is fundamental to climatology and hydrology, allowing the understanding and prediction of water demand in ecosystems and regions and helping to manage water resources worldwide more efficiently.

Artificial Intelligence (AI) can improve environmental modeling [18]. By adopting the concept of machine learning (ML) [19], we can determine logical rules capable of describing a complex system with minimal human intervention [20,21], including applications with meteorological variables. Machine learning techniques such as Artificial Neural Networks (ANN), Random Forest (RF), and Support Vector Machines (SVM) are widely used in various research areas, such as in hydrology and bioinformatics [13,22]. These tools can model complex, non-linear data, enabling accurate and reliable predictions in multiple applications [23].

Artificial Neural Networks (ANN) is a type of deep learning model that simulates the functioning of the human brain with layers of interconnected neurons. In hydrology, ANNs are widely used for flow forecasting and flood prediction [17]. On the other hand, Random Forests (RF) is an ensemble learning method that uses multiple decision trees to predict outcomes. They are used in several areas, most notably for predicting extreme events such as floods and checking water quality [24].

Support Vector Machines (SVM) is a model capable of mapping data into higher dimensional space, allowing for the separation of non-linear classes. They have also been used for flow and water quality prediction [25].

Despite all the advantages of machine learning techniques such as ANN, RF, and SVM, some points must be considered [13,22]. In particular, these models can be affected by so-called “overfitting,” which occurs when a model overfits training data and loses the ability to generalize to new data [23]. In addition, these models can require large amounts of data for training and can be computationally intensive [17].

Therefore, given the development and improvement of various methods of time series prediction [26], this work aims to apply different machine learning techniques to compare alternative strategies with the standard process, such as Artificial Neural Networks (ANN), Random Forests (RF) and Support Vector Machines (SVM), to obtain an analytical model that depends on fewer variables when compared to the standard Penman–Monteith method (PM-FAO) [27,28,29] and can cane evapotranspiration through an alternative technique.

Using control techniques in vegetation is something that has been introduced previously. In the 1950s, the pointers analyzed air temperature, relative humidity, and solar radiation, presenting a panoramic map of the variables [30].

Such learning techniques studied in this paper offer a variety of advantages over traditional methods of data analysis [13]. However, it is essential to consider the disadvantages and limitations of these features to ensure their correct and practical application [22].

2. Material and Methods

2.1. Study Area

The study was conducted at Fazenda Miranda, in the Brazilian Pantanal region, located at 15°43′53″ S and 56°04′18″ W, 15 km SSE of Cuiabá, Mato Grosso, Brazil, depicted in Figure 1, being at an altitude of 157 m above sea level and on flat terrain. It has forest-pastoral vegetation, known locally as Cerrado “campo dirty” [10,31].

The climate is characterized, according to the Köppen classification, as semi-humid tropical, with dry winters and humid summers [32,33,34], an average annual temperature of 26.5 °C, minimum temperatures around 5 °C, and maximum temperatures up to 41 °C [35]. The region has an average annual precipitation of 1420 mm, with a dry season between May and October [10,36].

The soil is nutrient-poor, locally known as Dystrophic Concretionary Soil, with a yellowish-red coloration [37,38]. It is pretty rocky, characterized by its high porosity, low water retention capacity, and very high post-rainfall drainage, which lasts between three and five days [10,31].

2.2. Data Acquisition

Temperature (T—°C), relative humidity (RH—%), global solar radiation index (RS—MJ/m²·d), and wind speed (V—m/s) were collected at 30 min intervals over ten years starting in 2009, with sample gaps of less than 5%.

These measurements were made using the following sensors: a thermohygrometer (HMP45AC, Vaisala Inc., Woburn, MA, USA) capable of measuring air temperature and relative humidity, installed 2 m above the ground; a pyranometer (LI200X, LI-COR Biosciences, Inc., Lincoln, NE, USA) used to measure global solar radiation, installed at the height of 5 m above the ground; and an anemometer (03101 R. M. Young Company, Traverse City, MI, USA) g the daily average of these parameters and the PM-FAO method, it was possible to determine the ETo daily for the studied period [10,31].

2.3. Penman–Monteith Method

The Penman–Monteith (PM) method is considered by the Food and Agriculture Organization of the United Nations (FAO) to be the standard method for estimating reference evapotranspiration (ETo), which is the evapotranspiration of a crop well adapted to the location, growing under the same conditions, used as standard evapotranspiration through crop coefficients. This method combines aerodynamic and energy balance components [39,40]. The following expression is used to estimate ETo by the Penman–Monteith method (FAO):

$$ETo=\frac{0.408\Delta (Rn-G)+\frac{\gamma 900U_2(e_s-e_a)}{T + 273}}{\Delta +\gamma (1 + 0.34U_2)}$$

(1)

where $\gamma$ is the slope of the vapor pressure curve concerning temperature (kPa °C⁻¹); Rn is the daily radiation balance (MJm⁻² day⁻¹); G is the daily total heat flux in the soil (MJm⁻² day⁻¹); is $\gamma$ the psychrometric coefficient (kPa °C⁻¹); U₂ is the wind speed at 2 m height (ms⁻¹); e_s is the vapor saturation pressure (kPa); e_a is the current vapor pressure (kPa); T is the average air temperature (°C).

When the Rn, G, U₂, and T values are measured at meteorological stations, it is necessary to calculate the values of $\Delta$, $\gamma$, e_s, and e_a.

To do this, the following relations are used:

$$\Delta =\frac{4098\left[0.6108·\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{17.27T}{T+237.3}\right)\right]}{{\left(T+237.3\right)}^2}$$

(2)

where exp is the base of the natural logarithm e, with e = 2.71828;

$$\gamma =0.665·{10}^{-3} \mathrm{P}\mathrm{a}\mathrm{t}\mathrm{m}$$

(3)

where Patm is the local atmospheric pressure (kPa), calculated based on the local altitude (z);

$$\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{m}=101.3{\left(\frac{293-0.0065z}{293}\right)}^{5.26}$$

(4)

with z being the local altitude.

The difference between e_s and e_a is called the saturation deficit. These values can be calculated using the following expressions:

$$e_s=0.6108\text{·}\exp \left(\frac{17.27T}{t+237.3}\right)$$

(5)

$$e_a=\frac{e_s·UR}{100}$$

(6)

RH is the relative humidity (%) the weather station provides.

2.4. Machine Learning Techniques

We used three machine learning techniques in this work (all supervised methods): Artificial Neural Networks (ANN), Random Forest (RF), and Vector Support Machine (SVM).

Different machine learning techniques, such as Artificial Neural Networks (ANN), Random Forests (RF), and Support Vector Machines (SVM), share the characteristic of being machine learning models, capable of identifying patterns in complex, non-linear data that allow accurate predictions to be made in various applications.

ANNs are computational mathematical algorithms based on the neural structures of intelligent organisms [41], as shown in Figure 2. They can reproduce the activity of the human brain. They can be used to create an artificial system capable of storing and extending knowledge based on experience (learning), providing analytical results for different applications [42]. The advantages of ANNs include handling non-linear problems, high dimensionality data, and real-time predictions. However, training can be time-consuming, and the network can suffer from overfitting with small data.

The ANN is a system modeling technique that can offer several advantages, including adaptation to the system being modeled, learning patterns in data without the need for labels or supervision, being functional in situations where there is sufficient labeled data or when you do not know what to look for in the data and can also be used to model systems that require reinforcement learning, where the model learns based on reward or punishment feedback. They can also be easily parallelized, meaning they can be run on specialized hardware to speed up processing time.

RF is a supervised learning algorithm that creates a “forest” of random decision trees [43]. It is a combination of decision trees, mainly trained with the bagging method, generating various learning models that increase the quality of the overall result [44]. The technique generates several decision trees and combines them to obtain a better and more stable forecast [45], as represented in Figure 3. On the other hand, RF techniques are helpful in classification and regression problems and tolerant of outliers and high-dimensional data, but have disadvantages of lack of interpretability and overfitting with small data.

Finally, we have SVM, shown in Figure 4, a classification algorithm for two data sets if the boundary between classes is linear [46]. In practice, however, we often deal with non-linear limits. One solution is to map the training set in its original (non-linear) space to a new higher dimensional space, called the linear feature space [47]. In turn, SVMs have high-class separation accuracy and handle high dimensionality data well but are sensitive to kernel choice and computationally intensive for large data sets.

We perform the same setup for all the experiments with these three techniques. For each method, we trained it with 70% of the variables collected in their original form (values contained between 2009 and 2016). We validated the remaining 30% (deals from 2017 to 2019), generating an analytical model capable of predicting the analyzed system with great precision, always respecting the increasing order of collection dates.

Choosing the most appropriate machine learning technique for a specific application depends on several factors, including the nature of the data, the goal of the analysis, and computational limitations. Each method has particularities and may be more advantageous for specific situations.

When using Artificial Neural Networks (ANN), different architectures were evaluated for each test to decide which was the best for the data analyzed. In common, all networks had two hidden layers. The number of neurons varied from 8 to 32 in each hidden layer. Three activation functions were evaluated: hyperbolic tangent, sigmoid, and rectified linear unit (relu). As optimizers, the RMSprop, and Adam algorithms were tested.

In our experiments, the RFs were configured to have 1000 trees in the forest with no depth limit, only two as the minimum number of samples needed to split an internal node, and only one as the minimum number of samples needed to be in a leaf node. This configuration makes the decision trees more flexible, despite the increased processing time. However, the amount of data in this investigation had little impact on the tests.

We performed a grid search with cross-validation for the SVM to set the main parameters: C, epsilon, and gamma. For each parameter, we checked three values: 0.1, 10, and 1000 for C; 0.01, 1, and 10 for epsilon; and 0.01, 1, and 10 for gamma. The best combination was used to train the method using the RBF kernel.

All tests were performed using Python in the Google Colaboratory (Colab) environment. We also used open source libraries such as Keras and Scikit. Learn, which have machine learning algorithms implemented, such as ANN, SVM, and RF.

2.5. Statistical Evaluation

After collecting in loco the parameters used (temperature (T (°C)), relative humidity (RH (%)), wind speed (WS (m/s)) and global solar radiation (GSR (MJ/m²·d)), making use of the three proposed methods ((ANN), (RF), and (SVM)), the values were introduced in the input layer of each routine, requiring as output the ETo values already known by PM-FAO [48,49,50].

Then, the combination between the input data (combinations of 1, 2, and 3 input variables) was performed to find the best variety that resulted in the highest level of approximation to the PM-FAO ETo [51]. In all the models used, statistical analyses of the predictions found by the different Machine Learning techniques were performed [44,45,46].

These procedures were carried out to compare the quality of the evapotranspiration prediction, always using the R statistical software [52,53,54,55]. Each model developed was used to predict the reference evapotranspiration through the validation sample group, thus obtaining the values predicted by each model. Three (3) metrics were calculated—R-squared, root mean squared error (RMSE), and mean absolute error (MAE)—which are used to evaluate and validate the efficiency of the results from each model [56].

The primary measure evaluated was the R-square (R²) (Equation (7)), which is Pearson’s linear correlation coefficient squared between the observed and predicted values [57,58].

$${\mathrm{R}}^2(y,\hat{y})=1-\frac{{\sum }_{i-1}^n(y_i-{\hat{y}}_i{)}^2}{{\sum }_{i-1}^n(y_i-\underset{¯}{y}{)}^2}$$

(7)

this is the coefficient of the squares of the differences between the observed values divided by the values estimated by the equation, divided by the sum of the squares of the differences between the observed values and the mean of the values observed in the results.

It is a value between zero and one, with higher values indicating a stronger correlation between the data [57,58,59].

The second measure calculated (Equation (8)) is the Root Mean Square Error (RMSE). It represents the average root mean square difference between the observed values and measures the average prediction error. The larger the RMSE, the worse the quality of the model [58].

$$\mathrm{RMSE}(y,\hat{y})=\sqrt{\frac{1}{n}{\sum }_{i-1}^n(y_i-{\hat{y}}_i{)}^2}$$

(8)

Finally, through Equation (9), the Mean Absolute Error (MAE) was used, which is the average absolute difference between the observed and predicted values. Similar to the RMSE, it is a measure of average prediction error, and the higher the MAE, the worse the model quality [58].

$$\mathrm{MAE}(y,\hat{y})=\frac{1}{n}{\sum }_{i-1}^n|y_i-{\hat{y}}_i|$$

(9)

3. Results and Discussion

3.1. Analysis of Micrometeorological Parameters

According to Figure 5a, the global solar radiation index variation can be observed in a restricted range from 3 to 30 (MJ/m²·d) over ten years, with an average of 17 (MJ/m²·d). This variation is due to the high frequency of clouds in the rainy seasons, which leads to a greater variety of this parameter in this period [60,61]. These changes in the solar radiation index are influenced by factors such as the amount of energy emitted by the sun, changes in brightness, and variations in the solar wind and magnetic field, which also affect local temperature, as well as the variability of local and seasonal parameters [60,61].

The temperature range ranged between 10 °C and 35 °C, shown in Figure 5b, with an average close to 27 °C. This is because the distance to seas and oceans can affect the temperature range, as these bodies of water serve as temperature moderators, smoothing out or amplifying temperature fluctuations [62,63,64]. These variations are more pronounced during the dry season, likely due to cold fronts coming from the southern region [65].

The relative humidity (Figure 5c), which is indirectly affected by solar radiation, also presents a wide range of values, between 10% and 100%, with an average of 80% during the period studied. Together with global solar radiation (GSR), relative humidity is crucial for calculating the reference evapotranspiration value (Eto) [66]. Thus, the relative humidity of the air is a fundamental element in the atmosphere because its presence, to a greater or lesser degree, influences temperatures, precipitation patterns, thermal sensation, and even our health [33,66].

We highlight that in the dry seasons of 2010, 2011, and 2012, the average value of the HR parameter reached values lower than 40%. During this period, we had less rainfall than in other seasons and periods of high temperatures. The variations in the relative humidity of the air were more pronounced during the dry periods. Still, compared with other areas of Cerrado, they were smaller due to the soil’s high drainage capacity and low water absorption power [31,66].

Contrary to the previously mentioned observations, we can observe in Figure 5d, wind in the region has a sporadic character [31,67,68], with amplitudes ranging from 0.5 m/s to 3.5 m/s, and an average around 1.3 m/s in the study region, which can be considered moderate [39,40,69].

We have the representation of the temporal evolution of the reference evapotranspiration in Figure 5e, which presents similar behavior in equal periods, thus reinforcing the idea that parameter modeling can occur through alternative tools [39,40].

3.2. Fitting the Models by Machine Learning Techniques

Forty-five different models were adjusted, combining the four initial input parameters, taken one by one, two by two, three by three, and four by four, and applied to the three chosen machine learning techniques (ANN, RF, and SVM). The primary sample was divided into two subsets: one for training, using 70% of the data, and another for validation, using the remaining 30%, maintaining the ascending order of collection dates.

Correlation graphs were generated for all combinations, as shown in Figure 6. The horizontal axis represents evapotranspiration calculated using the PM-FAO method, and the vertical axis represents evapotranspiration predicted by each technique. When a model correctly predicts an observation, the point will fall precisely on a straight line representing equality between the predicted and observed values. The further the issue is from this line, the worse the model’s prediction for that observation [70].

Table 1. Comparison of the best performances of the techniques used.

	Technique	Parameters	R-Squared	RMSE	MAE
1°	ANN	GSR + T + WS	0.9450	0.4329	0.1874
2°	ANN	GSR + RH + WS	0.9405	0.4398	0.1934
3°	ANN	GSR + T + RH	0.9320	0.4611	0.2126
4°	SVM	GSR + T + WS	0.8794	0.5443	0.2962
5°	SVM	GSR + RH + WS	0.8642	0.5625	0.3164
6°	RF	GSR + T + RH	0.8570	0.5505	0.3030
7°	SVM	GSR + T + RH	0.8471	0.5620	0.3158
8°	RF	GSR + T + WS	0.8261	0.5843	0.3414
9°	RF	GSR + RH + WS	0.8190	0.5885	0.3463

shows the three best parameter combinations for each technique based on R-square, RMSE, and MAE metrics, using the variables of global solar radiation (GSR), wind speed (WS), temperature (T), and relative humidity (RH) through their combinations to evaluate and select the best arrangements [57,58]. All varieties are shown in Appendix A.

In general, a high correlation means that as the observed value for a variable increases, the predicted value tends to increase as well, indicating an association between the observed and predicted values, but it does not measure error [57,58,71].

The table above shows the best techniques and their corresponding combinations, highlighting the three most efficient ones. We can see that the prediction generated by the Artificial Neural Network (ANN) technique, when used with a variety of three elements that all include the GSR parameter, shows excellent results based on the R-squared, RMSE, and MAE metrics, indicating that a smaller number of variables can represent the observed system. There is also information from the other two techniques used in this study, which show the results of using the same variables. Table 1 shows the best techniques and their observed combinations, highlighting the three most efficient ones. When used with various elements, including the RS parameter, the Artificial Neural Network (ANN) algorithm’s prediction shows excellent results based on the R-squared, RMSE, and MAE metrics, indicating that a smaller number of variables can represent the observed system.

Among the possible combinations, the RNA technique with the parameters GSR + T + WS stands out, with metrics of R² = 0.9450, RMSE = 0.4329, and MAE = 0.1874. This combination can accurately represent Reference Evapotranspiration even without the relative humidity parameter due to the high temperatures often recorded in the region, which tend to have a low influence on this climatic element [15,54,59], including the periods from 2010 to 2012, where the HR parameter presented low results.

The second best technique involves replacing the temperature with the relative humidity, still using the RNA, and also with three parameters, which results in a performance very close to the abovementioned. This model can therefore be considered equivalent to the previous one, even when analyzing periods of low relative humidity, that is, periods of prolonged droughts.

Third, we have the RNA technique using the GSR + T + RH parameters, which performs slightly less than the first two combinations. It is worth mentioning that this model does not use wind speed in its forecast. The variable WS shows a lower value when included, according to the performances presented in Appendix A. This is an exciting finding, as this technique can dispense with an anemometer, saving costs when setting up an ETo measurement station while still providing satisfactory results.

The inclusion of GSR in all combinations is expected, given that the region is characterized by annual temperature variations between 31 °C and 34 °C [60,61].

In addition to the possibility of savings with a data collection system, predicting a reliable model that depends on a smaller number of parameters can lead to an increase in the reliability of the model, as the aggregate error relative to the measurements is reduced [70].

It should be noted that air temperature and relative humidity can be measured with the same equipment, a thermo-hygrometer, so that no additional costs would be incurred, since wind speed is independent of the other parameters. The ANN technique using GSR + T + RH parameters stands out as it generates savings in the setup of the measurement station while still providing good results since the Artificial Neural Networks showed a result of approximately 95% efficiency when compared to the PM-FAO method, using only three variables.

The proposed methodology also allows us to understand the region’s energy flow and the future vegetation’s behavior. Based on this knowledge, we can plan and execute projects with less equipment and present results close to the PM-FAO method.

However, it is essential to note that choosing the most suitable modeling technique depends on the problem at hand and data availability. Although Artificial Neural Networks can offer several advantages for modeling systems, they also have some limitations. They can be considered “black boxes”, and it is difficult to understand how they came to a particular prediction or decision. If there is insufficient data, the ANN may not present an accurate result to the system because the training and the test are impaired.

It is a fact that they require many data to be adequately trained. If the data is limited or expensive to collect, it can be challenging to use this technique effectively. Another challenge is systems with limited resources because the method can be computationally intensive. It may be susceptible to overfitting, meaning the model adjusts training data very well and does not generalize well to new data. In our research, these facts were not harmful because we had many values collected over ten years.

So, considering some limitations is essential when choosing a modeling technique for a given application. Research analyzing other methods can be applied, aiming to mitigate the critical points of artificial neural networks, through associations of different styles.

4. Conclusions

This work was based on finding an alternative and effective method, based on Machine Learning techniques, capable of describing the Reference Evapotranspiration of a Brazilian region using fewer parameters when compared to the Penman–Monteith (FAO). The analysis using Machine Learning techniques showed that these procedures are highly efficient for modeling environmental systems because they can process an extensive database and find the best interaction between the parameters involved, with results higher than 98% compared to the standard method, using a smaller amount of variables.

Thus, through analytical models and various modeling techniques, it was possible to find an artificial intelligence technique capable of determining Reference Evapotranspiration (Eto) with a smaller number of initial parameters compared to Penman–Monteith (FAO) values since they are very close to those predicted by the standard method. These results are sufficient and noteworthy in describing the system in question.

The Artificial Neural Network technique proved the most efficient among the proposed plans. It can model the system using fewer parameters, an important fact since the data collection equipment may present processing failures. It provides patterns that other mathematical modeling techniques would take time to identify. Replacing the Penman–Monteith method (FAO) is a very effective alternative, as it can handle complex, non-linear systems, which can be challenging to model with other techniques. In addition, it is sufficient to overcome noise and uncertainty in the input data, leading to reasonable outputs, even when data is incomplete or inaccurate.

However, using Artificial Neural Networks to model reference evapotranspiration for the Pantanal region in question proved to be highly efficient and feasible because it requires fewer parameters to be collected, which can be captured by weather stations in loco, presenting a very satisfactory result from the point of view of economics and feasibility of the application of the technique, when compared to the standard Penman–Monteith (FAO) method.