Artificial Neural Networks versus Multiple Linear Regressions to Predict the Christiansen Uniformity Coefficient in Sprinkler Irrigation

Abstract

The Christiansen Uniformity Coefficient (CUC) describes the distribution of water in a sprinkler system. In this study, two types of models were developed to predict the Christiansen Uniformity Coefficient (CUC) of sprinkler irrigation systems: Artificial Neural Network (ANN), specifically the feed-forward neural networks, and multiple linear regression (MLR) models. The models were trained on a dataset of published research on the CUC of sprinkler irrigation systems, which included data on a variety of design, operating, and meteorological condition variables. In order to build the predictive model of CUC, 10 input parameters were used including sprinkler height (H), working pressure (P), nozzle diameter (D and d_a), sprinkler line spacing (S_L), sprinkler spacing (S_S), wind speed (WS), wind direction (WD), temperature (T), and relative humidity (RH). Fifty percent (50%) of the data was used to train ANN models and the remaining data for cross-validation (25%) and for testing (25%). Multiple linear regression models were built using the training data. Four statistical criteria were used to evaluate the model’s predictive quality: the correlation coefficient (R), the index of agreement (d), the root mean square error (RMSE), and the mean absolute error (MAE). Statistical analysis demonstrated that the best predictive ability was obtained when the models (ANN and MLR) utilized all the input variables. The results demonstrated that the accuracy of ANN models, predicting the CUC of sprinkler irrigation systems, is higher than that of the MLR ones. During the training stage, the ANN models were more accurate in predicting CUC than MLR, with higher R (0.999) and d (0.999) values and lower MAE (0.167) and RMSE (0.456) values. The R values of the MLR model fluctuated between 0.226 and 0.960, the d values oscillated from 0.174 to 0.979, the MAE values were in the range of 2.458% and 10.792%, and the RMSE values fluctuated from 2.923% to 13.393%. Furthermore, the study revealed that WS and WD are the most influential climatic parameters. The ANN model can be used to develop more accurate tools for predicting the CUC of sprinkler irrigation systems. This can help farmers to design and operate their irrigation systems more efficiently, which can save them time and money.

1. Introduction

The uniformity of water distribution in sprinkler irrigation is a measure of how evenly water is applied to a field. It is important to have a uniform water distribution to ensure that all of the crops receive the water they need. Numerous studies have concentrated on the definition of parameters that affect sprinkler uniformity systems [1,2,3]. A sprinkler water distribution pattern depends on system design parameters (sprinkler line spacing, riser height, and the diameter of the nozzle), operational factors (operating pressure), and climatic parameters (wind speed and wind direction) [4]. Sprinkler water application uniformity is classified as “low” when the Christiansen Uniformity Coefficient (CUC) is lower than 84% [5].

Sprinkler irrigation distribution patterns, characterized by the CUC, vary with the system used. The authors of [2] assumed that values of CUC between 70% and 85% could be expected in sprinkler irrigation systems. In a center pivot irrigation system, [6] reported that CUC values over 80% can be reached; [7] reported that the wind is the main environmental variable affecting sprinkler performance. Some studies found that meteorological parameters, such as humidity and temperature, influence water distribution patterns in sprinkler irrigation systems [6,8]. The authors of [9] conducted experiments to evaluate how water pressure, nozzle diameter, and wind speed affect the performance of sprinkler irrigation systems during irrigation. They found that nozzle diameter and operating pressure positively influence the CUC. Proportional increases in CUC are also expected when the nozzle diameter and operating pressure increase [9]. The Christiansen Uniformity Coefficient is affected by the distance separating nozzles [10]. The uniformity of water distribution in sprinkler irrigation systems generally decreases proportionally with an increase in the distance separating two successive sprinklers [6,11]. The nozzle type and diameter also affect water distribution uniformity, as well as the nozzle’s elevation above the ground [12].

The poor performance of sprinkler irrigation systems could significantly affect crop production [13,14,15]. The authors of [16] studied how the evenness of water distribution from sprinkler irrigation systems affects how much maize is produced per a particular area of land. Using regression analysis, the researchers found that the uniformity of water distribution from the sprinkler irrigation system explained between 28 and 77% of the variation in the maize grain yield. The uniformity of fertilizers application increases with sprinkler irrigation uniformity [17]. These authors also pinpointed that, in China, acceptable crop yields could be reached even if the CUC is equal to or above 75%. The authors of [18] evaluated the water application uniformity of a center pivot on winter wheat yield. They found that the average grain yield at the end span of the pivot was significantly lower (p < 0.05) since the application uniformity coefficient was 65%. Distributing water uniformly in a field can help farmers achieve maximum crop yields, better fruit quality, and higher values of water use efficiency (WUE) [19].

Nevertheless, carrying out experiments to evaluate the performance of a sprinkler irrigation system is a time and equipment consumer operation. The CUC could be estimated using artificial neural network (ANN) models. ANN is a computational model that mimics the way nerve cells work in the human brain. For nonlinear statistical data modeling, ANN is a very effective tool. This is because ANNs require fewer inputs and can accurately extract generalized relationships between input and output parameters without the need to understand the physical processes behind them [20]. As the amount of available data increases, accuracy increases [21].

Several research studies used the ANN technique to find solutions for a wide range of irrigation applications, for example, the prediction of wind drift and evaporation losses from sprinklers [22,23], the impact of wind on sprinkler distribution patterns [24], the distribution of water in humid bulbs under the drip [25], water infiltration in a surface irrigation system (furrow system) [26], irrigation schedule management [27], and the efficiency of drippers [26]. Nevertheless, there has been no actual study using ANN for modeling CUC in sprinkler irrigation. For that reason, this study aims to (1) explore and compare the capacity of the ANN and multiple linear regression (MLR) techniques to estimate the CUC with different input combinations, and (2) determine the most influential parameters on CUC modeling.

2. Materials and Methods

2.1. Experimental Data

The development of the artificial neural network and multiple linear regression models was based on experiments from published studies [7,28,29,30,31,32], as shown in

Table 1. Summary of the experimental data used in the model’s development.

H (m)	P (kPa)	D (mm)	da (mm)	SL (m)	SS (m)	T (°C)	RH (%)	WS (m s−1)	WD (°)	CUC (%)	Country	N	Reference
2.3	300	4.4	2.4	15	18	12, 31	31, 64	0.6, 6.5	23, 338	51, 94	Spain	21	[7]
2.3	205, 470	4.4, 5.2	2.4	15	15, 18	8, 27	11, 98	0.5, 8.8	3, 360	60, 99	Spain	11	[28]
2.0	350	4.8	2.4	18	18	8, 15	52, 81	0.9, 6.7	113, 315	72, 89	Spain	10	[29]
2.3	310, 460	4.0	2.4	15	15, 18	19, 35	23, 75	1.1, 7.8	0, 338	53, 96	Spain	12	[30]
2.3	240, 420	4.0, 4.8	2.4	15	15	5, 27	40, 86	0.4, 8.0	0, 338	75, 93	Spain	15	[31]
2.0	188, 392	4.0, 5.0	2.5	18	18	8, 31	29, 81	0.2, 7.6	88, 343	52, 93	Spain	25	[32]

H = riser height, P = operating pressure, D = main nozzle diameter, d_a = auxiliary nozzle diameter, S_L = spacing between sprinkler lines, S_S = spacing between sprinklers, T = air temperature, RH = relative humidity, WS = wind speed, WD = wind direction, CUC = Christiansen Uniformity Coefficient, and N = number of experiments for each research work.

. A total of 94 experiments were conducted to measure the CUC. Experiments were carried out with sprinkler irrigation systems under various climatic conditions, various spacings of sprinklers, and various operational parameters. Sprinklers were equipped with double nozzles and installed over the ground in a triangle, a rectangle, or a square arrangement.

The choice of input parameters was driven by their potential influence on the CUC, as well as their relevance to the physical processes inherent in sprinkler irrigation. The height of the sprinkler above the ground (H) was considered due to its impact on water distribution. Variations in sprinkler height can affect the uniformity of water application. The pressure at which the sprinkler system operates (P) is a fundamental factor influencing water distribution. Different working pressures can lead to variations in water uniformity. The size of the main nozzle orifice (D) directly affects the discharge rate. Variations in the main nozzle diameter can influence the distribution pattern and, consequently, the CUC. Similar to the main nozzle, the size of the auxiliary nozzle orifice (da) was considered as it plays a role in the overall water discharge. Changes in auxiliary nozzle diameter can impact water uniformity. The spacing between sprinkler lines (S_L) is a geometric factor that can influence water coverage. Different line spacings may impact the uniformity of the water application. The distance between individual sprinklers (S_S) is crucial for determining the overlap of water patterns. Variations in sprinkler spacing can affect water uniformity. Wind can cause water droplets to drift, affecting the distribution pattern. Wind speed (WS) was included as it may influence the CUC under varying weather conditions. The direction from which the wind is blowing can impact the distribution of water. Wind direction (WD) is a relevant factor, especially in open-field irrigation. Ambient temperature (T) was considered due to its potential influence on evaporation rates and water demand. Temperature variations may affect the uniformity of water application. Relative humidity (RH) levels in the air were included, considering their potential impact on evaporation rates. Variations in humidity may influence the CUC by affecting the efficiency of water application.

A brief description of the experiments used is presented below.

In the first experiment, [7] investigated the impact of spatial and temporal variations in water application on maize (Zea mays) yield under solid-set sprinkler irrigation. The sprinklers were positioned in an 18 × 15 m triangular pattern. The diameters of the main and the auxiliary nozzles were 4.4 mm, and 2.4 mm, respectively. The nozzles were operated under a constant pressure of 300 kPa. The findings of the experiment revealed that wind speed was the sole factor responsible for the variability in CUC. In the second experiment, [28] evaluated the impact of wind speed on the CUC under different combinations of sprinkler spacing (15 × 15 m and 18 × 15 m), main nozzle diameters (4.4 mm; 4.8 mm; and 5.2 mm), auxiliary nozzle diameter (2.4 mm), and operating pressures (220 kPa, 320 kPa, and 450 kPa). The findings revealed that CUC values improved linearly as the wind speed increased up to 2.0 m s⁻¹. However, beyond this point, CUC values began to decrease linearly as wind speed continued to increase. In the experiment conducted by [29], researchers carried out experiments to calibrate and validate the ADOR-Aspersion model for the main sprinkler triangular spacing (18 × 15 m), operating pressure (350 kPa), and nozzle diameters (main = 4.8 mm and auxiliary = 2.4 mm) used in the Montesnegros irrigation district (NE Spain). The calibrated and validated ADOR-Sprinkler model was used to characterize the effect of wind speed and direction on the CUC and water distribution resulting from a solid set. They proved that when the wind blows from a direction parallel to the sprinkler lines, the CUC is severely affected and reaches lower values than in other directions. The following experiment was carried out by [30]. The researchers calculated the CUC using the height of the irrigation water collected by a network of pluviometers installed within a solid-set sprinkler irrigation system arranged in a rectangular layout (18 m between sprinklers and 15 m among the lines). The diameters of the main and auxiliary nozzles were 4.0 and 2.4 mm, located at 2.3 m above the ground level. They found that wind speed is the primary factor influencing CUC values. In the fifth experiment [31], agricultural impact sprinklers under different combinations of meteorological conditions, operating pressure (240 kPa, 320 kPa, and 420 kPa), and nozzle diameter (main nozzle diameter = 4.0, 4.2 mm, and 4.8 mm; auxiliary nozzle diameter 2.4 mm) were evaluated. The CUC was evaluated for a rectangular 15 × 15 m solid-set sprinkler system arrangement. The experiment revealed that the wind speed is the most important factor affecting the value of CUC In the sixth experiment [32], different combinations of main and auxiliary nozzle diameters (4.0 + 2.5 mm, and 5.0 + 2.5 mm) and operating pressures (varying from 188 to 392 kPa) were tested using a solid-set system equipped with sprinklers arranged in an 18 m × 18 m square pattern (model RC130-BY) to study the impact of those combinations on CUC.

The following equation is used to calculate the CUC:

$$\mathrm{C}\mathrm{U}\mathrm{C}=100\times \left(1-\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{i}=25}\left|{\mathrm{I}\mathrm{D}}_{\mathrm{m}}-{\mathrm{I}\mathrm{D}}_{\mathrm{i}}\right|}{\mathrm{N}\times {\mathrm{I}\mathrm{D}}_{\mathrm{m}}}\right)$$

(1)

where N is number of observations; ID_m is average depth of gathered water; and ID_i is the depth of gathered water in each of the 25 pluviometers.

$${\mathrm{I}\mathrm{D}}_{\mathrm{m}}=\frac{\sum {\mathrm{I}\mathrm{D}}_{\mathrm{i}}}{\mathrm{n}}$$

(2)

The construction of ANN and MLR models was carried out by including the riser height (H), operating pressure (P), main nozzle diameter (D), auxiliary nozzle diameter (d_a), sprinkler line spacing (S_L), sprinkler spacing (S_S), wind speed (WS), wind direction (WD), air temperature (T), and relative humidity (RH). Ten (10) combinations of input variables for predicting CUC were designed to build ANN and MLR models (

Table 2. Combinations of input variables used in the ANN and MLR models.

ANN Model Nomination	MLR Model Nomination	Input Variables
ANN-1	MLR-1	H, P, D, da, SL, SS, T, RH, WS, WD
ANN-2	MLR-2	T, RH, WS, WD
ANN-3	MLR-3	WS, WD
ANN-4	MLR-4	T, RH
ANN-5	MLR-5	H, P, D, da, SL, SS
ANN-6	MLR-6	P, D, da
ANN-7	MLR-7	H, SL, SS
ANN-8	MLR-8	P, D, da, WS, WD
ANN-9	MLR-9	H, SL, SS, WS, WD
ANN-10	MLR-10	H, P, D, SL, WS, WD

H = riser height, P = operating pressure, D = main nozzle diameter, d_a = auxiliary nozzle diameter, S_L = spacing between sprinkler lines, S_S = spacing between sprinklers, T = air temperature, RH = relative humidity, WS = wind speed, and WD = wind direction.

). All of the available variables will be used as input variables for ANN-1 and MLR-1 to quantify how much each variable affects the prediction of CUC. The next combinations (numbers 2 to 4) were constructed to assess the model’s ability to predict CUC using only climate data and to determine the climate parameters that most influence CUC. From the fifth to the seventh combination, data related to design and operational conditions were utilized to build models to assess the models’ ability to predict CUC without relying on climatic data. For the last three combinations (eighth–tenth), WS and WD variables were combined with operational parameters (ANN-8 and MLR-8), with design parameters (ANN-9 and MLR-9), and with design and operational parameters (ANN-10 and MLR-10) to evaluate the influence of these two climatic data points on the CUC. More details of the input and output statistical parameters used to train and test the developed ANN and MLR models are shown in

Table 3. Statistical metrics of the input parameters used during the ANN training and testing phases.

Input Parameters
Statistical Metrics	H (m)	P (kPa)	D (mm)	da (mm)	SL (m)	SS (m)	T (°C)	RH (%)	WS (m s−1)	WD (°)	CUC (%)
Training input parameters
Xmean	2.2	284.1	4.4	2.2	16.3	16.5	18.4	58.7	3.3	233.1	82.3
Xmin	2.0	188.0	4.0	0.0	15.0	15.0	7.8	29.0	0.2	3.0	61.0
Xmax	2.3	452.0	5.0	2.5	18.0	18.0	31.0	91.0	7.8	343.0	97.0
SX	0.1	77.3	0.3	0.7	1.5	1.5	5.3	14.7	2.0	79.8	10.3
CV	0.07	0.27	0.07	0.34	0.09	0.09	0.29	0.25	0.60	0.34	0.12
CSX	−0.35	0.47	0.44	−2.65	0.35	0.00	−0.05	0.23	0.33	−0.71	−0.51
Testing input parameters
Xmean	2.2	305.1	4.6	2.4	15.7	17.7	19.2	55.1	3.0	198.9	81.3
Xmin	2.0	189.0	4.0	2.1	15.0	15.0	8.0	12.0	0.5	0.0	52.0
Xmax	2.3	367.0	5.0	2.9	18.0	18.0	30.0	83.0	7.0	336.0	96.0
SX	0.1	49.9	0.3	0.2	1.2	0.9	5.1	16.2	1.7	85.2	10.6
CV	0.07	0.16	0.07	0.07	0.08	0.05	0.27	0.29	0.57	0.43	0.13
CSX	−0.56	−1.34	−0.08	0.28	1.42	−2.60	−0.32	−0.49	0.54	−0.19	−1.19

H = riser height, P = operating pressure, D = main nozzle diameter, d_a = auxiliary nozzle diameter, S_L = spacing between sprinkler lines, S_S = spacing between sprinklers, T = air temperature, RH = relative humidity, WS = wind speed, WD = wind direction, and CUC = Christiansen Uniformity Coefficient. X_mean = mean value, X_min = minimum value, X_max = maximum value, S_X = standard deviation, CV = coefficient of variation, C_SX = skewness coefficient.

.

2.2. Artificial Neural Network (ANN)

ANN models are highly efficient data modeling tools that can be used to model multiple outputs simultaneously [33]. The ANN contains three neuron layers: the input layer (input variables), the hidden layer, and the output layer (output variables) [23] (Figure 1). Building an ANN model requires collecting data to train and test the model, choosing the best settings for the model, and evaluating the model on new data [33]. The optimal ANN model is created by changing the number of hidden layers, the number of neurons in each hidden layer, the learning algorithm, and the activation function. In this study, the ANN was trained using the backpropagation algorithm. Several algorithms could be used to accomplish the learning phase of the ANN. In the field of irrigation, the back-propagation (BP) algorithm is one of the commonly applied algorithms for training the neural network [34]. To find the best configuration for the ANN model, the researchers tested different numbers of hidden layers (1, 2, or 3), different numbers of neurons in each hidden layer (2 to 10), and different numbers of learning runs (1000 to 20,000). For the same level of accuracy, generalized feedforward (GFF) neural networks (Figure 1) used a reduced number of input variables to train the network. The authors of [35] used an iterative approach to determine the optimal number of processing elements (PE). This means they tried different numbers of neurons until they found one that gave them the best results. In this study, the hyperbolic tangent (tanh) transfer functions were considered to train the neural network since they are the most common transfer functions used in such kinds of studies [36].

To increase the capacity of the ANN models to generalize to new data, the researchers normalized the input and output data to an interval of 0–1 using the equation below:

$$\mathrm{X}=\frac{\mathrm{Y}-{\mathrm{Y}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{Y}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{Y}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}$$

(3)

where X is the normalized variable, Y is the observed value, and Y_min and Y_max are the minimum and maximum values, respectively.

In this case study, NeuroSolution software (version 5.0) was used to train and evaluate ANN models for predicting CUC [37]. The dataset (94 experiments) was divided randomly into training, cross-validation, and testing subsets. In total, 50% of these data were used for training (48 data points), 25% for the cross-validation process (23 data points), and 25% for testing (23 data points).

2.3. Multiple Linear Regressions (MLR)

Relationships between multiple variables can be linearly modeled using the technique of multiple linear regressions (MLR) [23,33]. The MLR technique allowed the researchers to identify the quantitative relationships between the independent and dependent variables [23]. The MLR model can be defined as follows:

$$\widehat{\mathrm{Y}}={\mathrm{b}}_{\mathrm{o}}+\sum {\mathrm{b}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}$$

(4)

where $\widehat{\mathrm{Y}}$ is the predicted value of the dependent variables, b_o is the intercept, X_i is the predictor variable, and b_i is the regression coefficient. Microsoft Excel software (version 10) was used to enact the MLR models. The MLR models were developed with the same experimental data used in the ANN models: 50% of the data points (randomly selected) were used to fit the MLR model and the remaining 50% used to test the model.

2.4. Criteria of Performance Evaluation

The accuracy of the ANN and MLR models was evaluated using four statistical criteria: correlation coefficient (R), index of agreement (d), mean absolute error (MAE), and root mean square error (RMSE). The selected criteria are defined by the following equations.

The correlation coefficient [38] is a statistical measure of the strength of the relationship between the experimental value and the predictive value. Values of R close to 1.0 mean good model performance.

$$\mathrm{R}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{X}}_{\mathrm{i}}-\overline{\mathrm{X}}\right)\times \left({\mathrm{Y}}_{\mathrm{i}}-\overline{\mathrm{Y}}\right)}{\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{X}}_{\mathrm{i}}-\overline{\mathrm{X}}\right)}^2\times {\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-\overline{\mathrm{Y}}\right)}^2}}$$

(5)

The index of agreement [39] is a statistical measure of the degree of agreement between two continuous variables varying from 0.0 to 1.0. Index of agreement values close to 1.0 indicate a good fit between measured and simulated CUC:

$$\mathrm{d}=1-\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{Y}}_{\mathrm{i}}-{\mathrm{X}}_{\mathrm{i}})}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\left|{\mathrm{Y}}_{\mathrm{i}}-\overline{\mathrm{X}}\right|+\left|\left({\mathrm{X}}_{\mathrm{i}}-\overline{\mathrm{X}}\right)\right|\right)}^2}$$

(6)

The root mean square error is used to measure the difference between observed values and predictive ones [40]. It measures the error value between observed and predictive data. The predictive quality of the used model increases when obtained values of RMSE decrease.

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{X}}_{\mathrm{i}}-{\mathrm{Y}}_{\mathrm{i}}\right)}^2}{\mathrm{n}}}$$

(7)

The mean absolute error (MAE) can be defined as the simplest measure of prediction accuracy. The MAE measures the average absolute discrepancy between measured and predicted values in a dataset. Values of MAE can range from 0 to infinity, and lower values indicate the greater predictive accuracy of the model [41].

$$\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{\mathrm{n}}\times \sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{X}}_{\mathrm{i}}-{\mathrm{Y}}_{\mathrm{i}}\right|$$

(8)

The Normalized Root Mean Square Error (NRMSE) relates the RMSE to the observed range of the variable. Thus, the NRMSE can be interpreted as a fraction of the overall range that is typically resolved by the model.

$$\mathrm{N}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\frac{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}{\overline{\mathrm{X}}}$$

(9)

where X_i is the measured value, Y_i is the estimated value, n is the number of observations, $\overline{\mathrm{X}}$ is the average measured value, and $\overline{\mathrm{Y}}$ is the average estimated value.

3. Results and Discussion

3.1. Effectiveness of ANN Models

Table 4. Statistical performance of ANN for the training and testing processes.

Model	Input Variables	HL	PE	Architecture	R	d	MAE (%)	RMSE (%)	NRMSE (%)
Training process
ANN-1	H, P, D, da, SL, SS, T, RH, WS, WD	2	5	10-2-5	0.998	0.999	0.333	0.645	0.008
ANN-2	T, RH, WS, WD	2	9	4-2-9	0.997	0.998	0.458	0.618	0.008
ANN-3	WS, WD	3	4	2-3-4	0.972	0.986	1.938	2.420	0.029
ANN-4	T, RH	2	9	2-2-9	0.876	0.928	3.646	5.010	0.061
ANN-5	H, P, D, da, SL, SS	2	9	6-2-9	0.749	0.849	5.729	7.181	0.087
ANN-6	P, D, da	2	5	3-2-5	0.733	0.835	5.375	6.949	0.084
ANN-7	H, SL, SS	1	10	3-1-10	0.607	0.487	7.896	8.812	0.107
ANN-8	P, D, da, WS, WD	3	6	5-3-6	0.989	0.993	1.260	1.660	0.020
ANN-9	H, SL, SS, WS, WD	2	7	5-2-7	0.988	0.994	1.227	1.613	0.020
ANN-10	H, P, D, SL, WS, WD	2	7	6-2-7	0.988	0.992	1.313	1.764	0.021
Testing process
ANN-1	H, P, D, da, SL, SS, T, RH, WS, WD	2	5	10-2-5	0.933	0.954	3.565	4.462	0.052
ANN-2	T, RH, WS, WD	2	9	4-2-9	0.935	0.964	3.391	3.833	0.045
ANN-3	WS, WD	3	4	2-3-4	0.912	0.928	2.957	3.730	0.043
ANN-4	T, RH	2	9	2-2-9	0.793	0.834	4.193	6.012	0.070
ANN-5	H, P, D, da, SL, SS	2	9	6-2-9	0.652	0.734	6.732	8.509	0.099
ANN-6	P, D, da	2	5	3-2-5	0.627	0.710	6.477	8.408	0.098
ANN-7	H, SL, SS	1	10	3-1-10	0.513	0.390	9.752	10.971	0.128
ANN-8	P, D, da, WS, WD	3	6	5-3-6	0.938	0.963	3.174	4.123	0.048
ANN-9	H, SL, SS, WS, WD	2	7	5-2-7	0.960	0.971	3.043	3.923	0.046
ANN-10	H, P, D, SL, WS, WD	2	7	6-2-7	0.949	0.969	3.174	3.984	0.046

HL = Hidden layer, PE = Processing elements, R = Coefficient of correlation, d = Index of agreement, MAE = Mean absolute error, RMSE = Root mean square error, NRMSE = Normalized root mean square error.

displayed the statistical performance of the ANN during the training period; the number of hidden layers and processing elements and the best neural network architecture is also mentioned. Figure 2 and Figure 3 show the comparison between the CUC value of the developed ANN model and the experimental value through a scatter plot of the training and the testing dataset. Each configuration of ANN models was run with 3 hidden layers (HL) and 10 neurons, or a processing element (PE) in each hidden layer. To better predict the CUC, the ANN architecture was assessed using the trial-error procedure. The optimal ANN configuration should minimize the MAE and the RMSE values and maximize the R and d values. As has been explained previously, for the first combination (ANN-1), all the input variables that were available for this study were used to construct the model. Using two HL and five PE in each hidden layer, the values of R and d statistical metrics were very close to 1.0 (0.998 and 0.999, respectively). These values suggest a strong correlation and agreement between the predicted and observed CUC values. This ANN architecture reached very low error values, with an MAE and RMSE of 0.333% and 0.645%, in that order. Therefore, the most appropriate ANN-1 architecture was 10-2-5. For ANN-1, we can say that increasing the number of HLs above two degraded the predictive quality of the model. The results indicate a high level of performance for the artificial neural network (ANN) architecture employed. As we can see, Figure 2 demonstrates that predicted values of the CUC follow a line of 1:1 during the training period, which indicates a good fit between the predicted and experimental values of the CUC for the ANN-1 model with a value of α₀ of 0.9957 (very close to one) and value of α₁ of 0.3222 (very close to zero). (fit line equation: $y={\alpha }_0·x+{\alpha }_1$). The high alignment between predicted and experimental values of CUC signifies a strong correspondence between the model’s predictions and the actual experimental data, indicating a high level of fitting accuracy.

To study the influence of meteorological variables on the ability of ANN to predict the CUC, three combinations of input parameters were used to construct ANN models, as described in Table 4 (ANN-2, ANN-3, and ANN-4). For the ANN-2 model, all the meteorological data were used (WS, WD, T, and RH). This input combination estimated very good CUC, reaching values of R and d of 0.997 and 0.998, respectively. Also, lower values of MAE and RMSE were obtained (0.475% and 0.763%, respectively). These low error values indicate that the model’s estimations are close to the actual observed values, emphasizing its reliability in capturing the complex relationships within the meteorological and CUC data. Performances of ANN-2 were very similar to those of ANN-1. Similar findings were obtained when only WS and WD were used for training ANN-3. Therefore, the R and d values were 0.972 and 0.986, respectively, using three HL and four PE in each hidden layer. These metrics suggest a good level of correlation and agreement between the predicted CUC values and the actual experimental data. The use of three HL and four PE in each layer reflects the complexity of the relationships being modeled. Nevertheless, we observed an increase in MAE and RMSE errors, reaching values of 1.938% and 2.420%, respectively. These findings suggest that, even with a reduced set of input variables, the model is effective in capturing the essential patterns in the data. However, when only T and RH inputs were used to train the ANN-4 model, we observed degradation in the performance criteria values. Values of R, d, MAE, and RMSE were 0.875, 0.930, 3.604, and 4.950, respectively, using two HL and nine PE in each hidden layer. Comparing the performance values of ANN-4 with ANN-2, the R and d values decreased by 12% and 7%, respectively, and the MAE and RMSE values increased by 659% and 549%, respectively. This indicates that relying exclusively on T and RH as input variables might not capture the complexity of the underlying patterns as effectively as when including additional meteorological variables. Therefore, we can pinpoint that within the meteorological variables, WS and WD had more influence than T and RH. In the absence of T and RH data, ANN models can be trained to predict the CUC of a sprinkle irrigation system. Overall, the degradation in performance metrics for the ANN-4 model suggests that T and RH alone may not provide an adequately robust basis for predicting the CUC in this context.

Design parameters (H, S_L, and S_S) and operational variables (P, D, and d_a) were used to develop ANN-5, ANN-6, and ANN-7 to predict the CUC and to study the most influential set of inputs. In the ANN-5, all six parameters were used and the values of R, d, MAE, and RMSE were 0.749, 0.851, 5.694, and 7.123, respectively, using two HL and nine PE in each hidden layer. Comparing the set of design and operational inputs (ANN-5) to the set of meteorological inputs (ANN-2), we observed a huge degradation in the performance of ANN models. The degradation in performance could be attributed to the absence of crucial meteorological information, such as WS and WD, in the ANN model (from 5 to 6). These variables likely play a substantial role in influencing the target variable (CUC) and contribute to the overall predictive accuracy of the model. Therefore, the values of R and d were reduced by 25% and 15%, respectively, and the values of MAE and RMSE were augmented by 1099% and 834%, respectively. As a conclusion, we can say that metrological data led to a better prediction of the CUC than design and operational data. The ANN-6 model was developed using only the operational data to predict CUC. The values of R, d, MAE, and RMSE were 0.732, 0.833, 5.419, and 7.006, respectively, using two HL and five PE in each hidden layer. Therefore, the best architecture of the ANN-3 model was found to be 3-2-5. When using only operational data, we found no significant difference in the accuracy of the ANN-5 and ANN-6 models. Nevertheless, when design parameters were used to predict CUC in the ANN-7 model, the performance criteria were changed negatively. The values of R, d, MAE, and RMSE were 0.607, 0.505, 7.817, and 8.738, respectively, using 1 HL and 10 PE in each hidden layer. Comparing the performance criteria values of ANN-7 to those of ANN-6, we pinpointed a decrease in R and d values by 17% and 39%, respectively, and a decrease in MAE and RMSE values by 44% and 25%.

Three ANN models (ANN-8 through ANN-10) were developed, mixing the WS and WD variables with (i) operational parameters (ANN-8), (ii) design parameters (ANN-9), and (iii) operational parameters with S_L. As we can see, there was a considerable improvement in ANN performance when WS and WD were taken into account with operational and design parameters. In the ANN-8, using operational parameters with WS and WD, the R, d, MAE, and RMSE values were 0.988, 0.993, 1.260, and 1.660, respectively, using three HL and six PE in each hidden layer. This improvement could be attributed to the fact that the combination of operational parameters with meteorological variables captures a more comprehensive set of factors influencing the target variable. Comparing the results of ANN-8 with those of ANN-6, the values of R and d improved by 35% and 19%, respectively, and the values of MAE and RMSE diminished by 77% and 76%, respectively. The same conclusion was reached when the WS and WD variables were mixed with the design data in the ANN-9 model. Therefore, when using two HL and seven PE in each hidden layer, the values of R, d, MAE, and RMSE were 0.987, 0.994, 1.227, and 1.613, respectively. This finding indicates that the inclusion of wind-related variables in conjunction with design parameters contributes to a better understanding of the underlying patterns in the data. Comparing the results of ANN-9 with those of ANN-7, the values of R and d were augmented by 63% and 97%, respectively, and the values of MAE and RMSE declined by 84% and 82%, respectively. Even when we integrate the S_L data into the operational parameters and we mix it with WS and WD data in ANN-10, the accuracy of the model in forecasting the CUC always remained high. Thus, the values of R, d, MAE, and RMSE were 0.987, 0.992, 1.313, and 1.764, respectively, using two HL and seven PE in each hidden layer. The positive impact on performance observed in this case further supports the notion that the inclusion of meteorological variables, specifically WS and WD, alongside operational parameters and an additional variable like S_L, leads to improved predictive accuracy. As a conclusion, we can pinpoint the importance of WS and WD data in predicting the CUC. In all ANN models developed using WS and WD parameters, the performance criteria were very high, with values of R and d close to 1.0 and values of MAE and RMSE less than 2.0%. This underscores the importance of considering a comprehensive set of input variables that capture the diverse factors influencing the system under study. Even when the WS and WD parameters were used alone, the performance criteria of the developed model were very high. In conclusion, even if we can only use WS and WD to predict CUC, the integration of design and operational parameters in the CUC prediction of sprinkler irrigation systems is fundamental and very important. The nozzle size plays an important role in the determination of water droplet diameter. The effect of WS and WD is even more pronounced when the water droplets are finer, thereby directly impacting the CUC.

3.2. Effectiveness of MLR Models

Table 5. Equations derived from the MLR models.

Model	Model Equation
MLR-1	$\mathrm{C}\mathrm{U}\mathrm{C}=178.9-0.01\mathrm{P}-2.4\mathrm{D}+1.4{\mathrm{d}}_{\mathrm{a}}-2.3{\mathrm{S}}_{\mathrm{L}}-2.2{\mathrm{S}}_{\mathrm{S}}+0.2\mathrm{T}+0.1\mathrm{R}\mathrm{H}-3.0\mathrm{W}\mathrm{S}-0.04\mathrm{W}\mathrm{D}$
MLR-2	$\mathrm{C}\mathrm{U}\mathrm{C}=102.3-0.21\mathrm{T}+0.05\mathrm{R}\mathrm{H}-3.02\mathrm{W}\mathrm{S}-0.04\mathrm{W}\mathrm{D}$
MLR-3	$\mathrm{C}\mathrm{U}\mathrm{C}=101.6-3.07\mathrm{W}\mathrm{S}-0.04\mathrm{W}\mathrm{D}$
MLR-4	$\mathrm{C}\mathrm{U}\mathrm{C}=79.3-0.18\mathrm{T}+0.11\mathrm{R}\mathrm{H}$
MLR-5	$\mathrm{C}\mathrm{U}\mathrm{C}=130.3-0.01\mathrm{P}+4.74\mathrm{D}+1.57{\mathrm{d}}_{\mathrm{a}}-4.14{\mathrm{S}}_{\mathrm{L}}-0.15{\mathrm{S}}_{\mathrm{S}}$
MLR-6	$\mathrm{C}\mathrm{U}\mathrm{C}=28.6+0.03\mathrm{P}+11.04\mathrm{D}-1.17{\mathrm{d}}_{\mathrm{a}}$
MLR-7	$\mathrm{C}\mathrm{U}\mathrm{C}=151.0-4.12{\mathrm{S}}_{\mathrm{L}}-0.11{\mathrm{S}}_{\mathrm{S}}$
MLR-8	$\mathrm{C}\mathrm{U}\mathrm{C}=84.6+0.02\mathrm{P}+3.08\mathrm{D}-1.53{\mathrm{d}}_{\mathrm{a}}-3.06\mathrm{W}\mathrm{S}-0.03\mathrm{W}\mathrm{D}$
MLR-9	$\mathrm{C}\mathrm{U}\mathrm{C}=162.3-1.74{\mathrm{S}}_{\mathrm{L}}-2.05{\mathrm{S}}_{\mathrm{S}}-3.05\mathrm{W}\mathrm{S}-0.03\mathrm{W}\mathrm{D}$
MLR-10	$\mathrm{C}\mathrm{U}\mathrm{C}=179.0-0.01\mathrm{P}-3.12\mathrm{D}-3.92{\mathrm{S}}_{\mathrm{L}}-3.21\mathrm{W}\mathrm{S}-0.02\mathrm{W}\mathrm{D}$

P = operating pressure, D = main nozzle diameter, d_a = auxiliary nozzle diameter, S_L = spacing between sprinkler lines, S_S = spacing between sprinklers, T = air temperature, RH = relative humidity, WS = wind speed, WD = wind direction, and CUC = Christiansen Uniformity Coefficient.

shows the equations obtained from the MLR models for the 10 constructed combinations.

Table 6. Standard error of regression coefficients, t statistic, and probability of independent variables for MLR models.

Model		Intercept	H (m)	P (kPa)	D (mm)	da (mm)	SL (m)	SS (m)	T (°C)	RH (%)	WS (m s−1)	WD (°)
MLR-1	SE	15.2	0.0	0.01	2.04	0.79	0.74	0.72	0.15	0.05	0.28	0.01
	t-stat	11.7	65.5 × 103	−1.73	−1.18	1.81	−3.10	−3.03	1.20	1.07	−10.85	−4.75
	p-value	3.2 × 10−14	-	-	0.25	0.08	3.6 × 10−3	4.4 × 10−3	0.24	0.29	3.3 × 10−13	2.9 × 10−5
MLR-2	SE	9.4	-	-	-	-	-	-	0.25	0.09	0.53	0.01
	t-stat	10.9	-	-	-	-	-	-	−0.84	0.54	−5.56	−2.86
	p-value	5.8 × 10−14	-	-	-	-	-	-	0.41	0.59	1.1 × 10−6	0.01
MLR-3	SE	2.92	-	-	-	-	-	-	-	-	0.52	0.01
	t-stat	34.74	-	-	-	-	-	-	-	-	−5.89	−2.97
	p-value	3.8 × 10−34	-	-	-	-	-	-	-	-	4.5 × 10−7	4.8 × 10−3
MLR-4	SE	14.3	-	-	-	-	-	-	0.39	0.14	-	-
	t-stat	5.54	-	-	-	-	-	-	−0.46	0.75	-	-
	p-value	1.5 × 10−6	-	-	-	-	-	-	0.65	0.46	-	-
MLR-5	SE	33.4	0.0	0.02	4.72	1.99	1.64	1.56	-	-	-	-
	t-stat	3.9	65.5 × 103	−0.44	1.00	0.79	−2.52	−0.10	-	-	-	-
	p-value	3.3 × 10−4	-	-	0.32	0.43	0.02	9.62	-	-	-	-
MLR-6	SE	27.1	-	0.02	5.25	2.17	-	-	-	-	-	-
	t-stat	1.1	-	1.23	2.10	−0.54	-	-	-	-	-	-
	p-value	0.3	-	0.18	0.04	0.59	-	-	-	-	-	-
MLR-7	SE	13.9	0.0	-		-	1.54	1.51	-	-	-	-
	t-stat	10.8	65.5 × 103	-	-	-	−2.68	−0.07	-	-	-	-
	p-value	4.1 × 10−14	-	-	-	-	-	0.94	-	-	-	-
MLR-8	SE	19.61	-	0.01	3.59	1.41	-	-	-	-	0.53	0.01
	t-stat	4.31	-	1.44	0.86	−1.08	-	-	-	-	−5.81	−2.45
	p-value	9.6 ×10−3	-	0.16	0.39	0.28	-	-	-	-	7.4 × 10−7	0.02
MLR-9	SE	5.78	0.0	-	-	-	0.73	0.70	-	-	0.28	0.01
	t-stat	28.08	65.5 × 103	-	-	-	−2.39	−2.94	-	-	−10.96	−4.09
	p-value	2.7 × 10−29	-	-	-	-	-	0.01	-	-	4.9 × 10−14	1.8 × 10−4
MLR-10	SE	14.85	0.0	0.01	2.06	-	0.41	-	-	-	0.30	0.01
	t-stat	12.05	65.5 × 103	−1.23	−1.51	-	−9.49	-	-	-	−10.66	−3.13
	p-value	3.2 × 10−15	-	-	0.14	-	5.2 × 10−12	-	-	-	1.6 × 10−13	3.1 × 10−3

SE = standard error, t-stat = t statistic, p-value = probability, H = riser height, P = operating pressure, D = main nozzle diameter, d_a = auxiliary nozzle diameter, S_L = spacing between sprinkler lines, S_S = spacing between sprinklers, T = air temperature, RH = relative humidity, WS = wind speed, WD = wind direction, CUC = Christiansen Uniformity Coefficient.

shows values of the standard error (SE), the t-stat, and the p-values of the independent variables in each equation listed in Table 5. At the 95% confidence level, WS variables reached t-stat values ranging from −10.66 to −5.56 for all MLR models. Negative t-stat values for WS show that the more the wind speed increases, the more the CUC decreases. The negative correlation indicated by the t-stat values suggests that wind speed is a relevant factor in explaining variations in CUC. Absolute values of t-stat were greater than 1.99 and the p-value of WS in Table 6 is very small. This shows that WS is the most important variable in calculating CUC. The observation that the absolute values of the t-statistics for wind speed (WS) are greater than 1.99, coupled with a very small p-value in Table 6, indicates a high level of statistical significance for WS in the calculation of CUC. The authors of [24] demonstrated that for a specific sprinkler with a specific nozzle size operating at the optimal operating pressure under field conditions, the resulting water distribution depends on the magnitude of the wind speed. The SE of the WS coefficients ranged from 0.28 to 0.53 with an average value of 0.40, which implies that there is some variability in the precision of the estimates across different models or datasets.

Regarding the WD variable, the t-stat values for all MLR models ranged from −4.75 to −2.45. This shows that the CUC decreases as the wind direction increases. The absolute value of the t-stat was greater than 1.99. The p-value for WD in Table 6 is too small. The mean SE of the WD coefficient was 0.01. This result shows that WD is also an important variable in the calculation of CUC [5]. WD plays a crucial role in determining the spatial distribution of water droplets ejected from the nozzles of the sprinkler system. The WD directly affects the trajectory of water droplets, and when the wind aligns with the direction of water application, it can lead to more uniform water coverage [29,32]. Conversely, if the wind is perpendicular or at an angle to the direction of water discharge, it can cause the water droplets to drift, leading to uneven distribution and potentially impacting the CUC [29,33]. Concerning the rest of meteorological parameters (T and RH), the absolute values of t-stat ranged from 0.46 to 1.20, lower than 1.99, and p-values varied from 0.24 to 0.65 (p > 0.05). Therefore, T and RH variables had no significance in the models. These findings imply that T and RH, as included in the models, do not have a statistically significant impact on predicting the CUC. The lack of statistical significance suggests that changes in T and RH are not associated with significant variations in the CUC within the context of the studied models.

Considering the group of operational parameters (P, D, d_a), no single variable other than the main nozzle diameter of the MLR-6 model is important for the calculation of CUC. For the MLR-6, the value of the t-stat for the D variable is 2.10, which is greater than 1.99. The p-value for D is rather small, at 0.04. This suggests that changes in the main nozzle diameter are associated with a significant impact on CUC within the context of the MLR-6 model. Regarding the remaining operating parameters, the absolute values of t-stat are low, between 0.44 and 1.81, and less than 1.99. The p-value is greater than 0.05 and ranges from 0.08 to 0.59. This result indicates that operational variables do not play a role in the calculation of the CUC. The lack of statistical significance for the other operational variables implies that variations in P and d_a are not associated with significant changes in the CUC within the considered model.

Regarding the design parameters and in all MLR models, the absolute values of the t-stat of the S_L variable ranged from 2.39 to 9.49, greater than 1.99. As is shown in Table 6, the p-values of the S_L variable are very small. This result indicates that the S_L variable has a great influence on the estimation of CUC. This finding suggests that changes in this variable are associated with a significant impact on the calculated CUC. Concerning the S_S parameter, absolute values of t-stat were greater than 1.99 in only MLR-1 and MLR-9 models, with values of 3.03 and 2.94, respectively. The p-values of the S_S variable were 0.004 for MLR-1 and 0.01 for MLR-9 (Table 6). The average value of the SE of the S_S coefficients was 0.71. The performance of the MLR model during the training and testing processes is shown in

Table 7. Statistical performance of MLR models during the training and testing processes.

Model	Input Variables	Training Process					Testing Process
		R	d	MAE (%)	RMSE (%)	NRMSE (%)	R	d	MAE (%)	RMSE (%)	NRMSE (%)
MLR-1	H, P, D, da, SL, SS, T, RH, WS, WD	0.960	0.979	2.458	2.923	0.036	0.943	0.971	3.333	4.252	0.049
MLR-2	T, RH, WS, WD	0.806	0.884	4.854	6.005	0.073	0.744	0.838	7.667	9.950	0.116
MLR-3	WS, WD	0.788	0.870	5.271	6.280	0.076	0.780	0.860	8.833	10.851	0.126
MLR-4	T, RH	0.226	0.174	8.479	10.003	0.122	0.498	0.571	8.583	11.303	0.131
MLR-5	H, P, D, da, SL, SS	0.629	0.732	6.729	7.993	0.097	0.525	0.624	5.167	6.481	0.075
MLR-6	P, D, da	0.608	0.704	6.854	8.151	0.099	0.324	0.450	6.250	7.724	0.090
MLR-7	H, SL, SS	0.365	0.367	8.438	9.627	0.117	0.374	0.399	10.125	12.061	0.140
MLR-8	P, D, da, WS, WD	0.808	0.327	10.792	13.393	0.163	0.862	0.919	5.208	6.668	0.078
MLR-9	H, SL, SS, WS, WD	0.948	0.972	2.667	3.279	0.040	0.921	0.772	8.125	9.585	0.111
MLR-10	H, P, D, SL, WS, WD	0.942	0.968	2.917	3.512	0.043	0.913	0.946	4.875	5.601	0.065

R = Coefficient of correlation, d = Index of agreement, MAE = Mean absolute error, RMSE = Root mean square error, NRMSE = Normalized root mean square error.

.

3.3. ANN vs. MLR Models

Figure 2 and Figure 3 show the scatter plots of the measured and estimated CUC of ANN and MLR during the model learning and testing phase. In general, the results predicted by the ANN model were more satisfactory than those predicted by the MLR model during the training process. The ANN models predicted CUC values that were more closely clustered around the 1:1 line than the MLR models. The values of α₀ for the ANN models were closer to one than those of the MLR models. Regarding the values of α₁, they were closer to zero for the ANN models than those obtained for the MLR models. The good predictive quality of ANN, compared to MLR models, is reflected in the value of R² for the ANN model, which is higher than that for the MLR model. The values of R² range from 0.368 to 0.998 for the ANN model and from 0.051 to 0.922 for the MLR model. The ANN model and MLR model produced similar results during the model fitting phase. Table 4 and Table 7 also show that when the ANN model is used to estimate the CUC, the values of R and d are high, and the values the MAE and RMSE are low. Comparing the performance of MLR-1 to ANN-1 models, using all variables, we pinpointed a reduction of R and d values by 4% and 2%, respectively, and an increase in MAE and RMSE values by 638% and 353%, respectively. When comparing the performance of the ANN and MLR models, the same trends were observed. There was always a decrease in R and d values, with percentages ranging from 4.0 to 74% and 2.0 to 81%, respectively, and an increase in MAE and RMSE values with percentages ranging from 17 to 3056% and 11 to 1277%, respectively. The MLR-7 model shows an increase in the d value by 45% and a small decrease in MAE and RMSE values by 13% and 8%, respectively. This suggests a strong correlation between the predicted CUC values by the ANN model and the actual experimental values. In comparison to the ANN model, the accuracy of the MLR models in estimating the CUC was found to be inferior. Consequently, the ANN model proved to be an effective and robust tool for modeling the CUC.

Several factors could contribute to this observed difference in performance between the ANN and MLR modeling approaches. (i) ANN models are fundamentally more flexible and capable of capturing non-linear relationships compared to linear regression models like MLR. (ii) ANNs are adept at learning hierarchical and abstract representations of features, allowing them to automatically extract relevant information from the input data. (iii) If certain input variables have non-linear or complex relationships with the target variable, ANNs might perform better in capturing these nuances. MLR assumes linear relationships, and if the true relationships are non-linear, it may lead to suboptimal predictions. (iv) ANNs have the ability to adapt to the complexity of the data by adjusting the number of hidden layers and nodes. This flexibility enables them to model intricate relationships effectively. MLR, being a simpler model, might struggle to capture the full complexity of the underlying data.

4. Conclusions

ANN and MLR techniques were employed to forecast the CUC of a sprinkler system. ANN and MLR used ten different input variables related to climate, sprinkler design, and operating parameters to predict the CUC. A type of artificial neural network called a generalized feedforward was utilized to train the ANN model. The correlation coefficient (R), index of agreement (d), mean absolute error (MAE), root mean square error (RMSE), and the normalized RMSE (NRMSE) were used to evaluate the predictive quality of the models. The following conclusions can be drawn:

• ANN models yielded results that were more consistent with the experimental CUC values than the MLR models;
• The model, including all input factors, performed very well in the forecasting of the CUC; however, it was not possible to determine which parameters have a significant impact on CUC prediction;
• It was found that models that included only the WS and WD input variables performed very well in the prediction of the CUC;
• When only T and RH were used to predict the CUC, the prediction quality for the model was degraded;
• The predictive quality of models including only operational and design variables was medium. The integration of WS and WD into those models enhanced their predictive quality;
• The MLR evaluation showed that the WS and WD variables are significantly associated with the CUC (p < 0.05), which was in good agreement with the results of the ANN analysis.

This study demonstrated that ANN is a powerful technique for forecasting the CUC of sprinkler irrigation systems. Furthermore, the trained and tested model could help irrigation managers to avoid high wind speeds and maximize the CUC.