1 Introduction

Crop reference evapotranspiration (ETo), a vital element in the hydrological cycle, plays a significant role in the effective use of existing water resources and planning sustainable water management. An immediate and certain consequence of global warming is an increase in temperature (Salman et al. 2017; Pour et al. 2019). Increasing temperature will alter ETo, which will have significant impacts on water resources. For water resource management, planning, and development, accurate estimation of ETo has become crucial due to the global warming and climate change impact studies (Muhammad et al. 2021). It is vitally important for agriculture and irrigation to estimate the amount of evapotranspiration. An incorrect measurement of evapotranspiration may lead to excessive plant water requirements, nutrient leaching, and groundwater pollution (Kisi et al. 2021). Several meteorological variables including global solar radiation (Rs), ambient temperature (Tmax and Tmin), relative humidity (RH), and wind speed (U2) influence the evapotranspiration process. Huang et al. (2019) predicted ETo using meteorological variables of maximum temperature (Tmax), minimum temperature (Tmin), relative humidity (RH), wind speed (U), and solar radiation (Rs). Majhi and Naidu (2021) estimated ETo using data of maximum temperature (Tmax), minimum temperature (Tmin), sunshine hours (SH), minimum wind speed (Umin), relative humidity (RH), and cumulative pan evaporation (Ep). Maroufpoor et al. (2020) used maximum temperature (Tmax), minimum temperature (Tmin), relative humidity (RH), sunshine hours (SH), average wind speed (Umean), and precipitation (P) to estimate ETo. The FAO-56 PM model which has been used in many areas to estimate ETo, however, is limited by the fact that it requires many climatic variables. The required instruments for this model can be difficult and expensive to install and maintain (Exner-Kittridge and Rains 2010; Valiantzas 2015). Therefore, this model is rarely used especially in developed countries (Shiri 2017; Yamaç and Todorovic 2020; Koudahe et al. 2018; Djaman et al. 2019).

The ETo of plants can be determined by equations like FAO Penman, FAO Penman–Monteith, and Blaney–Criddle (Feng et al. 2017; Chen et al. 2019a, 2019b). Even though these methods perform well in most cases, they cannot handle nonlinear complexities well and require a vast amount of input data, which may not always be available. In recent years, artificial intelligence (AI) and soft computing techniques have gained significant attention in a variety of engineering and medical fields. A literature review of time series, empirical, and data-driven modeling techniques was conducted to simulate ETo (Huang et al. 2019; Fan et al. 2018; Nourani et al. 2019; Yu et al. 2020; Sharafi and Ghaleni 2021; Douna et al. 2021; Keshtegar et al. 2022; Mirzania et al. 2023; Majumdar et al. 2023). There have already been several conventional artificial neural network (ANN) algorithm modeling techniques that have proven to be successful in modeling ETo (Achite et al. 2022; Gocić and Amiri 2021; Nawandar et al. 2021; Abrishami et al. 2019; Hashemi and Sepaskhah 2020). Antonopoulos and Antonopoulos (2017) predicted daily ETo in northern Greece and found that the ANN model has a more accurate prediction than other models. Ferreira et al. (2019) investigated the performance of ANN and support vector machine (SVM) models in estimating daily ETo in Brazil and concluded that the results of the ANN model are better than the other one.

According to a recent study, Elbeltagi et al. (2022) modeled the ETo in Ludhiana (i.e., sub-humid climate) located in Northern India using five hybrid ANN-based models including artificial neural networks-additive regression (ANN-AR), ANN-random forest (ANN-RF), ANN-REPtree, ANN-M5Pruning tree (ANN-M5P), and ANN-bagging models. Finally, the ANN-M5P model was selected as the best model based on the performance criteria values of MAE = 0.730 mm/day, RMSE = 0.959 mm/day, NSE = 0.779, and WI = 0.935. Maroufpoor et al. (2020) estimated ETo using the standalone ANN model, hybrid artificial neural network-gray wolf optimization (ANN-GWO) model, and least square support vector regression (LS-SVR) models in Iran. Finally, the hybrid ANN-GWO model was selected as the best model with the values of MAE = 0.279, SI = 0.077, and R2 = 0.981 at the testing phase.

Ahmed et al. (2022) used the convolutional neural networks and gated recurrent unit (CNN-GRU) model to predict ETo for a week ahead and obtained R = 0.993 and RMSE = 0.087 for the superior model at the validation stage. Zhao et al. (2019) applied a set of global climate models (GCMs) to determine ETo at the seasonal and monthly scales. The COOT-ANN hybrid model has not been used to estimate ETo in Australia.

In the field of crop reference evapotranspiration, several studies have been conducted using machine learning (ML) techniques (Mokari et al. 2021; Kaya et al. 2021; Niaghi et al. 2021; Dias et al. 2021; Chen et al. 2020; Üneş et al. 2020; Farias et al. 2020). New algorithms are being developed globally to improve accuracy, reduce errors, and lower costs. Consequently, many researchers employ such algorithms when developing models in a variety of fields. These researchers strive to measure the efficiency and effectiveness of their models on an industrial and global scale.

In recent years, the hybrid intelligent models have been developed to enhance the performance of the standalone AI models. Bio-inspired optimization algorithms draw significant attention for developing hybrid models in hydrological studies like estimating ETo (Ahmadi et al. 2021). Seifi and Riahi (2020) proposed three hybrid models, namely, least square support vector machine-gamma test (LSSVM-GT), artificial neural network-gamma test (ANN-GT), and adaptive neuro fuzzy inference system-gamma test (ANFIS-GT) for predicting daily ETo at arid areas of Iran. The results of this study showed that the hybrid LSSVM-GT model provides higher accuracy than the others. Maroufpoor et al. (2020) estimated evapotranspiration using ANN, artificial neural network-gray wolf optimization (ANN-GWO), and least square support vector regression (LS-SVR) models in Iran; the result showed the high ability of the ANN-GWO model than ANN and LS-SVR models. Gao et al. (2021) used three hybrid models, namely, artificial neural network-bat algorithm (BA-ANN), Cuckoo Search algorithm-artificial neural network (CSA-ANN), and whale optimization algorithm (WOA-ANN) to model daily ETo with limited inputs and found that the hybrid WOA-ANN generated better estimations than the BA-ANN and CSA-ANN. Yang et al. (2022) showed that a calibrating anomaly improves daily reference crop evapotranspiration forecasts with two models Australian Community Climate and Earth-System Simulator G2 version (ACCESS-G2) and Seasonally Coherent Calibration (SCC) in Australia.

The Coot bird natural life model (COOT) algorithm is an innovative algorithm, which has not been used in hydrological problems; however, it has successfully been used in other different fields. Mawgoud et al. (2022) investigated an effective hybrid approach based on the arithmetic optimization algorithm and sine cosine algorithm for the integration of battery energy storage system in distribution networks by the (COOT). They have used various models such as sine–cosine approach (SCA), arithmetic optimization algorithm (AOA), dynamic programming (DP), and coupled models of arithmetic optimization algorithm-sine cosine approach (AOA-SCA) models. Mostafa et al. (2022) used the modified COOT algorithm (mCOOT) based on two techniques opposition-based learning (OBL) and orthogonal learning for dimensionality reduction.

In this study, the COOT algorithm was used to improve the accuracy and effectiveness of artificial neural networks. Although hybrid models may be able to perform better than single models, it is usually an exchange between the training time and the performance of the model. In this study, to enhance the efficiency and easier training of the model, a hybrid predictive model based on the ANN algorithm and the COOT method (COOT-ANN) was developed. The developed hybrid COOT-ANN model would help to resolve some weaknesses of ANN, such as being trapped in local optimal solutions, which could result in higher accuracy and lower error rates.

According to the author’s best knowledge, there are no studies on the application of hybrid COOT-ANN model in water resource fields. The purpose of this study is to develop and evaluate the performance of the hybrid COOT-ANN model in predicting ETo and to compare its performance with the standalone intelligent ANN model. In this research, a novel hybrid model was developed and used to predict ETo in various climate stations in Australia. The results were compared with a simple machine learning model which has been used previously to predict ETo in a single station or climate. This research makes an important contribution to assessing crop reference evapotranspiration under different climates. The findings of the present study could help in recommendations for modeling crop reference evapotranspiration in different climate in Australia. to planning sustainable water resources management.

2 Materials and methods

2.1 Case study and data used

Data required for this research include daily minimum and maximum temperatures (Tmin (°C), Tmax (°C)) and crop reference evapotranspiration (ETo (mm/day)) which were collected from January 1, 2010, to December 31, 2021, from seven meteorological sites in Australia. The temperature data were obtained from the Australian Meteorological Agency (BOM) (http://www.bom.gov.au), while the ETo data were obtained from both meteorological station records and satellite measurements. Seven meteorological stations in seven different states of Australia, namely, Leinster Aero, Charleville Aero, Stawell Aerodrome, Tennant Creek Airport, Woomera Aerodrome, Butlers Gorge, and Mudgee Airport, were used in this study (Fig. 1). The stations with complete data and no missing or noisy data were selected. The information related to each station is presented in Table 1. Table 2 shows the statistical characteristics of the daily meteorological data in different stations. For training and testing of the models, 80% and 20% of daily data were used, respectively. A time series diagram of the observed daily ETo data for the selected stations is presented in Fig. 2. The time series plot of crop reference evapotranspiration for seven stations (Fig. 2) indicates that the Butlers Gorge station has the lowest crop reference evapotranspiration rate and the Tennant Creek Airport station has the highest crop reference evapotranspiration rate. Australia is a large continent in the Oceania region spanning 35° of latitude and, as such, has several climate zones. Australia’s northern regions have a tropical climate and large amounts of evapotranspiration are caused by the hot, humid weather. In contrast, in the southern part, there are distinct seasons with dry summers and wet winters; coastal areas also tend to have a low evapotranspiration rate due to their proximity to large water sources.

Fig. 1
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Location of the meteorological stations in seven states in Australia

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Table 1 Characteristics of the stations
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Table 2 Statistical characteristics of the meteorological data
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Fig. 2
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Observed daily crop reference evapotranspiration time series at the studied stations (2010–2021)

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2.2 Artificial neural network

The function of an ANN is to receive, process, and generate information much like that of biological neuronal networks. In ANNs, repetition is the key to learning. When different data is entered into the learning algorithm numerous times, the difference in the amount of data can be detected by the changes that occur in the weight and bias vectors. The back-propagation of error, also called back propagation (BP), is one of the most widely used methods for training the ANN. In this algorithm, each iteration involves two steps: the first step is a feed-forward motion, which multiplies inputs with weights and sums them with bias vectors. Feed-forward motion involves measuring the error values after obtaining the output values and comparing them with the targets. So, by finding out about error values related to the weights and biases, the algorithm moves on to the next step, which is back-propagation, and modifies the parameters based on the error values. Iterations (feed-forward and back-propagation) are performed until the predicted output is very close to the desired output (target). In order to get a better cost function topology, it is better to scale the data into the given interval before training an ANN model. An optimization algorithm may not be able to find the optimal values if the data is not scaled into the given scale. A network can learn optimal parameters more quickly when data is scaled within a given interval (Nourani 2017).

2.3 Coot bird natural life model

The Coot bird natural life model (COOT) algorithm simulated the different collective behaviors of Coots, a small water bird in the rail family, using a metaheuristic optimization algorithm. On the surface of the water, coots make regular and irregular movements; this behavior is ultimately aimed at moving towards food or to a specific location (Naruei and Keynia 2021). Four different movement patterns can be observed on the water surface of a coot group: random movement, chain movement, adjusting position according to leader, and leader movement (Naruei and Keynia 2021). The COOT algorithm is implemented using these four movements’ behaviors. The specific implementation procedure is as follows (Naruei and Keynia 2021): creation of the population is randomly initialized by according to Eq. (1):

$$\mathrm{CootPos}(i)=\operatorname{rand}\ \left(1,d\right)\times \left(\mathrm{ub}-\mathrm{lb}\right)+\mathrm{lb}$$
(1)

where CootPos(i) represents the position of the coot, d is the number of variables or dimensions of the problem, ub and lb represent the upper and the lower bound of the search space, respectively.

$$\mathrm{ub}=\left[{\mathrm{ub}}_1,{\mathrm{ub}}_2,\ldots, {\mathrm{ub}}_d\right],\mathrm{lb}=\left[\ {\mathrm{lb}}_1,{\mathrm{lb}}_2,\ldots, {\mathrm{lb}}_d\right]$$
(2)

Following the initialization of the population, the coot’s position is updated according to four movement patterns.

2.3.1 Random movement

The position Q for this movement is first randomly initialized by using Eq. (3):

$$Q=\operatorname{rand}\left(1,d\right)\times \left(\mathrm{ub}-\mathrm{lb}\right)+\mathrm{lb}$$
(3)

The position is updated according to Eq. (4) in order to avoid being trapped in a local optimum:

$$\mathrm{CootPos}(i)=\mathrm{CootPos}(i)+\mathrm{A}\times {R}_2\times \left(Q-\mathrm{CootPos}(i)\right)$$
(4)

where R2 is a random number in the interval [0, 1], and A is calculated as Eq. (5):

$$A=1-L\times \left(\frac{1}{\mathrm{Iter}}\right)$$
(5)

where Iter is the maximum number of iterations and L is the current number of iterations.

2.3.2 Chain movement

In order to implement the chain movement, the average position of the two coot birds can be calculated using Eq. (6):

$$\mathrm{CootPos}(i)=\frac{\mathrm{CootPos}\left(i-1\right)+\mathrm{CootPos}(i)}{2}$$
(6)

In this case, CootPos(i − 1) indicates the location of the second coot.

2.3.3 Adjusting position according to the leader

During each group, a coot bird’s position updates based on the position of the leader; therefore, the follower moves towards the leader. The leader is selected using Eq. (7):

$$K=1+\left(i\ \mathrm{MOD}\ \mathrm{NL}\right)$$
(7)

where K is represented as the number of the leader index, i is the number of the coot bird follower, and NL is the number of leaders.

A coot’s position is updated according to Eq. (8) during this movement:

$$\mathrm{CootPos}(i)=\mathrm{LeaderPos}(k)+2\times {R}_1\times \mathrm{Cos}\left(2 R\pi \right)\times \left(\ \mathrm{LeaderPos}(k)-\kern0.5em \mathrm{CootPos}(i)\right)$$
(8)

Using CootPos(i) as the current position of the coot bird, the LeaderPos(k) represents the position of the selected leader, R1 is a random number in the interval [0, 1], and R is a random number in the interval [−1, 1].

2.3.4 Leader movement

Leader positions are updated using Eq. (9) based on the leap from local optimal positions to global optimal positions:

$$\mathrm{LeaderPos}(i)=\left\{\begin{array}{c}B\times {B}_3\times \mathrm{Cos}\left(2\pi R\right)\times \\ {}\left(g\mathrm{Best}-\mathrm{LeaderPos}(i)\right)+g\mathrm{Best}\kern3em {R}_4<0.5\\ {}B\times {B}_3\times \mathrm{Cos}\left(2\pi R\right)\times \\ {}\left(g\mathrm{Best}-\mathrm{LeaderPos}(i)\right)+g\mathrm{Best}\kern3.5em {R}_4\ge 0.5\end{array}\right.\kern0.5em$$
(9)

where gBest refers to the best position that can be found, R3 and R4 are the random numbers between the interval [0, 1], and R is the random number between the interval [−1, 1]. B is determined from Eq. (10):

$$B=2-L\times \left(\frac{1}{\mathrm{Iter}}\right)$$
(10)

2.4 Hybrid model (COOT-ANN)

In the COOT-ANN model, the optimization COOT algorithm is applied to determine the best and optimum values of the ANN’s parameters and so increase its ability in accurately predicting different problems. Hybrid algorithms have many advantages, including their flexibility of implementation, unconditional formulation of problems, and absence of derivatives. When a mathematical function reaches certain fitness between COOT and ANN, then hybrid models (i.e., COOT-ANN) stop, or when iterations reach the maximum number, then the model stops. This approach allows the models to reach their maximum capabilities, and then the new hybrid model can have advantages of the ANN and optimization (Mohammadi et al. 2020). Table 3 shows the parameters used in the modeling process. Figure 3 illustrates a flowchart of the COOT-ANN model used throughout the research. Figure 3 presents a summary of the COOT-ANN hybrid algorithm; it depicts how the hybrid algorithm process involves the combining of two or more algorithms to solve a particular problem, with the goal of setting the optimal parameters to reduce error and improve accuracy. Figure 4 presents the Pseudo code of the COOT algorithm and an overview of parameters which has been used and updated, as well as the performance of the COOT algorithm with the best configuration during learning process.

Table 3 Parameters used for training the ANN and COOT-ANN models
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Fig. 3
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Flowchart of the COOT-ANN model

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Fig. 4
figure 4

Pseudo code of the COOT bird natural life model

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2.5 Performance evaluation statistics

The predictive models are evaluated using six performance criteria, namely, correlation coefficient (R), root mean squared error (RMSE), Nash-Sutcliffe efficiency (NSE), RMSE-observation standard deviation ratio (RSR), Scatter Index (SI), and mean absolute error (MAE):

Correlation coefficient (R)

$$R=\frac{\sum_{i=1}^N\left(\mathrm{EO}-\overline{\mathrm{E}0}\right)\left(\mathrm{EP}-\overline{\mathrm{E}\mathrm{P}}\right)}{\sqrt{\sum_{i=1}^N{\left(\mathrm{EO}-\overline{\mathrm{E}\mathrm{O}}\right)}^2.{\sum}_{i=1}^N{\left(\mathrm{EP}-\overline{\mathrm{E}\mathrm{P}}\right)}^2\kern0.5em }}-1\le R\le 1$$
(11)

Root mean square error (RMSE)

$$\mathrm{RMSE}=\sqrt{\frac{1}{N}{\sum}_{i=1}^N{\left(\mathrm{EP}-\mathrm{EO}\right)}^2\ 0\le \mathrm{RMSE}\le \infty }$$
(12)

Nash–Sutcliffe Efficiency Coefficient (NSE)

$$\mathrm{NSE}=1-\left[\frac{\sum_{i=1}^N{\left(\mathrm{EO}-\mathrm{EP}\right)}^2}{\sum_{i=1}^N{\left(\mathrm{EO}-\overline{\mathrm{EO}}\right)}^2}\right]-\infty <\mathrm{NSE}\le 1$$
(13)

RMSE-observation standard deviation ratio (RSR)

$$\mathrm{RSR}=\frac{\sqrt{\sum_{i=1}^N{\left(\mathrm{EO}-\mathrm{EP}\right)}^2}}{\sqrt{\sum_{i=1}^N{\left(\mathrm{EO}-\overline{\mathrm{EO}}\right)}^2}}\kern0.5em 0\le \mathrm{RSR}\le \infty$$
(14)

Scatter Index (SI)

$$\mathrm{SI}=\frac{\sqrt{\frac{1}{N}{\sum}_{i=1}^N{\left(\mathrm{EP}-\mathrm{EO}\right)}^2}}{\overline{\mathrm{EO}}}0\le \mathrm{SI}\le \infty$$
(15)

Mean absolute error (MAE)

$$\mathrm{MAE}=\frac{1}{N}{\sum}_{i=1}^N\left(\mathrm{EP}-\mathrm{EO}\right)0\le \mathrm{MAE}\le \infty$$
(16)

where EO and EP are the observed and predicted ETo values, N denotes the number of observations, and \(\overline{\mathrm{EO}}\) and \(\overline{\mathrm{EP}}\) are the means of the observed and predicted ETo. Table 4 indicates classification of a model’s performance based on the NSE and RSR criteria values.

Table 4 Model performance based on NSE and RSR criteria
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3 Application and results

In order to achieve an optimal model in a data-driven modeling approach, input variables are being fed into the models one by one. In this study, daily Tmax (°C) and Tmin (°C) variables in all the stations were used as the inputs of the classical ANN and hybrid COOT-ANN models. Initially, the back propagation feed forward algorithm was used in the classical ANN model, while in the second step, the Coot bird natural life algorithm was coupled with ANN to increase the model’s preciseness. Figure 5 illustrates a summary of the methodology used in this study. The performance of developed models was evaluated by the Taylor diagrams and the statistical metrics including R, RMSE, NSE, RSR, SI, and MAE. Table 5 shows the models’ performance based on the six statistical measures in the training and test periods for all the seven stations for ANN and COOT-ANN models.

Fig. 5
figure 5

Schematic diagram of the proposed methodology

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Table 5 Performance metrics of the models in the training and testing process for ANN and COOT-ANN modes
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The statistical analysis shows that the COOT-ANN model is able to predict the daily ETo values more accurately than the ANN models in the testing period for all the 7 stations based on the lower values of the RMSE, RSR, SI, and MAE and higher values of the R and NSE criteria. This proves that the COOT algorithm was successful in determining the optimal weight and bias of the ANN model in all the stations. However, comparing the model’s performances at the stations reveals that both the models do not perform satisfactory at Tennant Creek Airport station (e.g., the NSE values are 0.515 and 0.581for the ANN and COOT-ANN models, respectively). Table 5 shows that the proposed COOT-ANN has detected the best solution with the maximum cost, high accuracy, and less error with substantial progress compared to ANN model.

The scatter and time series plots of the observed and predicted daily ETo of the ANN and COOT-ANN models, for all the seven stations, are presented in Figs. 6 and 7. In the scatter plot, a series of dots represents the values of two numerical variables. The horizontal axis represents the values of one variable, while the vertical axis represents the values of the second variable. An analysis of scatter plots is used to examine relationships between variables. Using a scatter plot, the dots indicate not only the values but also patterns that can be observed based on the data. A trend line was added to the scatter plot to display the mathematically best fit to the data when analyzing predictive or correlational relationships between variables. Dots with a clearly clustered pattern, or those that follow a curve or trend line closely, are considered to have a strong relationship. A time series plot is useful for determining how the data has trended over time as well as if the data points are random or show any patterns over time. The time series plot is also useful for comparing the results of different models. In analyzing the results of the time series graph, the lower error shows that the observed values are well predicted by the model and there are the fewer outlier’s data. Comparing the time series plots of these two figures indicates that in overall, the COOT-ANN model is significantly more accurate than the ANN model in estimating ETo especially the extreme (minimum and maximum) values in all stations (except the Tennant Creek Airport station at which both of the models do not show satisfactory performance). Also, according to the scatter plots, the predicted data of the COOT-ANN model (Leinster Aero station (R2 = 0.819), Charleville Aero (R2 = 0.825), Stawell Aerodrome (R2 = 0.828), Tennant Creek Airport (R2 = 0.594), Woomera Aerodrome (R2 = 0.862), Butlers Gorge (R2 = 0.715), and Mudgee Airport (R2 = 0.813)) have higher R2 values, and show less scatter than those of the ANN values (Leinster Aero station (R2 = 0.806), Charleville Aero (R2 = 0.815), Stawell Aerodrome (R2 = 0.817), Tennant Creek Airport (R2 = 0.575), Woomera Aerodrome (R2 = 0.855), Butlers Gorge (R2 = 0.705), and Mudgee Airport (R2 = 0.798)) (Figs. 6 and 7).

Fig. 6
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Scatter and time series plots of the ANN model during the testing phase

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Fig. 7
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Scatter and time series plots of the COOT-ANN model during the testing phase

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In Fig. 8, the Taylor diagram can be used to compare the models. In the next step, the ANN and COOT-ANN models were compared with each other using the Taylor diagrams (Taylor 2001) in the testing phase (Fig. 8). As can be seen in the Taylor diagram, it is possible to visualize how closely model estimations match the observations (Taylor 2001). In a Taylor diagram, standard deviation (SD), root mean square deviation (RMSD), and correlation coefficient (R) are summarized in one diagram suitable for illustrating the effectiveness of predictive tools based on polar diagram. As the azimuth angle represents the coefficient of correlation, the radial location from the beginning represents the standard deviation proportion. Each model is measured by its distance from the observed point along the horizontal axis. The distance between a model and the observed point is a measure of its accuracy. Due to this, the closest point to the observed point performs the best. The Taylor diagram (Fig. 8) shows the prediction made by COOT-ANN algorithm and ANN model at all seven stations, where the red contours are representative of correlation coefficients, blue contours are representative of RMSD errors, and black contours are indicative of standard deviations. Figure 8 illustrates that the COOT-ANN models are more accurate and higher performance than the ANN based on the distance of the COOT-ANN point from the observed point for all stations ((Leinster Aero station (RANN = 0.898, RCOOT-ANN = 0.905; RMSEANN = 1.254, RMSECOOT-ANN = 1.205), Charleville Aero (RANN = 0.903, RCOOT-ANN = 0.908; RMSEANN = 1.090, RMSECOOT-ANN = 1.040), Stawell Aerodrome (RANN = 0.904, RCOOT-ANN = 0.910; RMSEANN = 0.983, RMSECOOT-ANN = 0.944), Tennant Creek Airport (RANN = 0.759, RCOOT-ANN = 0.771; RMSEANN = 1.513, RMSECOOT-ANN = 1.406), Woomera Aerodrome (RANN = 0.925, RCOOT-ANN = 0.929; RMSEANN = 1.199, RMSECOOT-ANN = 1.160), Butlers Gorge (RANN = 0.840, RCOOT-ANN = 0.846; RMSEANN = 0.765, RMSECOOT-ANN = 0.750), Mudgee Airport (RANN = 0.894, RCOOT-ANN = 0.902; RMSEANN = 1.014, RMSECOOT-ANN = 0.972))).

Fig. 8
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Taylor diagrams of the models in the testing period for seven stations

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4 Discussion

The results of hybrid and standalone models for simulating ETo (Table 5) indicated that the hybrid COOT-ANN model performed more successfully than the ANN models using only the Tmax and Tmin input variables. Moreover, the statistical indices (including correlation coefficient (R), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), RMSE-observation standard deviation ratio (RSR), Scatter Index (SI), and mean absolute error (MAE)) along with the Taylor diagrams are used to determine whether there are significant differences between COOT-ANN and ANN models in a statistical sense. This is in accordance with other similar studies by Silva et al. (2020), Adnan et al. (2021), and Sabzevari and Eslamian (2022). The current study indicates that the hybrid model produced a very promising and encouraging results. The performance criteria indicated that the R criterion of the hybrid model increased by up to 0.803%, the RMSE reduced by up to 4.127%, the NSE increased by up to 3.359%, the RSR reduced by up to 4.072%, the SI reduced by up to 4.148%, and the MAE reduced by up to 3.665% compared to the ANN model at the testing period. Overall, based on the statistical and visual measures, it is found that in this study similar to the other previous studies (e.g., Seifi and Riahi 2020; and Maroufpoor et al. 2020), the hybrid intelligent models outperform the simple standalone models due to the good performance of the heuristic optimization algorithms (i.e., the COOT algorithm in this study). The COOT-ANN algorithm has several advantages over the ANN model, including simplicity, ease of implementation, and few parameters. To the best of our knowledge, no study has applied this algorithm to prediction of daily crop reference evapotranspiration in Australia. The COOT optimization algorithm is primarily designed to solve continuous optimization problems. Therefore, to address discrete optimization problems, such as crop reference evapotranspiration on a daily basis, the COOT-ANN operators and parameters must be modified.

It should be noted that ETo processes, like any natural processes, may exhibit both linear and nonlinear behaviors over time. In cases where a time series includes both linear and nonlinear patterns, combining linear and nonlinear models could produce better results. The superior efficiency of this method is achieved by training a black box ensemble method that uses historical observations to learn what weights should be applied to the component parts of the model based on different patterns of historical observations.

There have been very few studies conducted in Australia on prediction of evapotranspiration, which rely primarily on simple artificial intelligence models (machine learning) or numerical methods related to meteorology (Falamarzi et al. 2014; Perera et al. 2014). But in the present study, a novel hybrid machine learning model (COOT-ANN) was employed to determine the accuracy of the hybrid model compared with the simple model (ANN). One research gap which has been studied in this research was comparison of the results of a simple machine learning model with a completely new and developed hybrid model. In previous studies, they mainly used a lot of weather input variables to predict evapotranspiration, while in this research, only two input variables (Tmax and Tmin) were used to predict crop reference evapotranspiration in this research, the basic COOT and ANN algorithms were successfully combined to develop a novel hybrid meta-heuristic algorithm (COOT-ANN) with better all-around search performance for global optimization. To assess the performance of the developed COOT-ANN algorithm, the maximum iteration, population size, and learning rate were set as 1000, 500, and 0.001, respectively. Experimental results demonstrated that COOT-ANN performs better than pure model solution accuracy, convergence speed, stability, and local optima avoidance.

The accuracy of the applied ANN-based models differs significantly at the different selected sites based on two factors of statistical characteristics (such as the skewness, kurtosis, coefficient of variation, and average) of the training data and then the generated weights of the models in various sites. To achieve the best results from the hybrid models, it is important that the input data have lower skewness, kurtosis, coefficient of variation, and higher mean value (Rajaee and Shahabi 2015; Kisi 2005).

5 Conclusion

One of the most important components of the hydrological cycle is crop reference evapotranspiration. A wide range of the hydrological and water resource processes are affected by crop reference evapotranspiration. It is therefore imperative to quantify this process for the sustainable management of water resources. The Penman–Monteith method is a standard method for estimating the crop reference evapotranspiration, ETo (Allen et al. 1998); however, this model is not capable of estimating the ET0 in regions with limited meteorological data as the method requires a large number of variables. In this study, a new optimization algorithm, i.e., COOT bird natural life algorithm hybridized with the artificial neural network (COOT-ANN model) was developed and evaluated for the prediction of daily ET0 using just two meteorological data (Tmax and Tmin) from seven stations in different states of Australia. The COOT-ANN model was compared with the classical ANN model using statistical indices such as correlation coefficient (R), root-mean-squared error (RMSE), Nash–Sutcliffe efficiency (NSE), RMSE-observation standard deviation ratio (RSR), Scatter Index (SI), mean absolute error (MAE), and Taylor diagrams. Several factors are important in the development of new algorithms, including accuracy enhancements, error reductions, model simplicity, ease of model development, run time reduction, and less computations. Researchers from all over the world continuously introduce more new optimizer models, which are inspired by a variety of topics and sources. However, the new algorithms are not able to optimize the coefficients of the artificial neural network models and are largely trapped in their local optimal points. Therefore, only a limited number of them can effectively optimize ANN coefficients in comparison to classical methods. The results of this study proved that the COOT-ANN model was more accurate and efficient than the classical ANN model in estimating daily ETo at all seven stations. It is recommended that the COOT-ANN model to be applied and evaluated in modeling other hydrological variables in Australia and other parts of the world. Moreover, since only seven sites were selected in this study, it is recommended that the validity of the developed hybrid model to be investigated in different locations with different agro-climatic conditions. Also, hybridization of the COOT algorithm with other intelligent standalone techniques and examining the developed models in prediction problems is recommended. However, a significant portion of future research can be devoted to minimizing uncertainties and improving optimization performance.