The study site is an irrigated perimeter located in the Iffou region, in the department of M’Bahiakro in central-eastern Ivory Coast. It lies between longitudes 4018’ and 4020’ West and latitudes 7026’ and 7031’ North and includes the first hydro-agricultural inflatable dam in Ivory Coast [10] ( Figure 1 ).

The rice-growing perimeter, divided into two sectors, covers an area of 450 ha and is 9 km long and 1 km wide [11] . The irrigation system is gravity-fed, with open canals and two water storage towers. The hydraulic districts are grouped into twelve zones or blocks, with six (06) blocks per sector and two (02) water pumping stations ( Figure 2 ). The soils in the irrigated perimeter generally have a clay-loam texture (70%) with hydraulic conductivities varying between 10⁻⁴ and 10⁻³ m/s [10] . The drainage porosity of the soil in the irrigated area varies between 40% and 60%. The highest soil porosity values are observed to the north of the irrigated perimeter, with a maximum average value of around 54%.

The physico-chemical parameters of the water, in particular pH, temperature (T˚C) and electrical conductivity (EC), were measured in situ using a HANNA type HI 9828 portable multiparameter calibrated according to the study season. At the same time, three water samples for nitrogen analysis (nitrites, nitrates, and ammonium) were taken the following day between five and six in the morning. The water samples were taken using three 500 mL polyethylene bottles immersed directly in the water of the dam by means of a rope. Each bottle was rinsed three times with the water to be sampled, then filled to the brim and hermetically sealed before being stored in a cooler. The bottles containing the water samples were stored at a temperature of 4˚C, as indicated by a thermometer in the cooler. This temperature of 4˚C maintained in the cooler with the aid of ice accumulators is useful for maintaining the stability of all the nitrogenous elements in the bottles. The preserved water samples were taken to the laboratory on the day of sampling. In the laboratory, nitrogen parameters were analyzed within twelve hours of sampling, using a flame molecular absorption spectrophotometer in accordance with [20] .

Soil samples were taken using an auger at 20 points spread across the study area. The twenty samples were taken to a depth between 0 and 30 cm of soil, equivalent to the organic horizon of the soil. This is also the soil zone (0 - 30 cm) most affected by farming activities [2] . The collection and transport of the twenty soil samples and the analysis of their chemical parameters were carried out in accordance with the protocols defined by the French standards agency (AFNOR). The methods used to analyze the chemical parameters of the soil are summarized in Table 1 . In addition, physical data relating to fine soil density (Dt), moisture at field capacity (θcc), moisture at the permanent wilting point (θpf), and the cultural coefficient (Kc) at the irrigated perimeter were determined in situ.

As part of this study, the values of the crop coefficient (Kc) were established in accordance with the recommendations of [21] , as presented in FAO Irrigation and Drainage Paper No. 56. These coefficients take into account the different phenological stages of rice, distributed over twelve ten-day periods covering the entire vegetative cycle. Their variation allows for a more accurate estimation of water requirements at each stage of crop development and contributes to optimal irrigation planning ( Table 2 ).

To obtain data on rice yields as a function of increasing doses of nitrogen, a 1-hectare area of the irrigated perimeter was set aside for rice experimentation with increasing doses of nitrogen. The experiment was carried out over 6 months (from August to February 2020) and consisted first of dividing the 1 hectare of the cleared area into four equal parts of 2500 m² each. Three parts of the 1 hectare each received 70 kg∙N∙ha⁻¹, 90 kg∙N∙ha⁻¹ and 140 kg∙N∙ha⁻¹ of nitrogen, and the fourth part, with no nitrogen applied, was chosen as the control according to Figure 3 below. The doses applied were the result of a synthesis of the nitrogen doses applied to each experimental plot from the soil preparation phase through to the ripening of the rice plants. The rice yield results obtained following this experiment were reported for each quantity of nitrogen applied to the different plots.

This section briefly describes artificial neural networks, in particular, the gradient back-propagation neural network (BPNN) used in this study.

Artificial neural network methods are statistical tools used to estimate complex phenomena. Their specificity lies in their ability to estimate non-linear systems. These methods are particularly suitable for reproducing agricultural and hydrological processes and are therefore widely used in soil and water quality modelling ( [17] [18] ). Among neural models, Multilayer Perceptron models are inspired by the architecture and functioning of the human brain, where the brain attempts to learn from signals coming from its environment in order to provide a response or action to be taken. This type of neural model is made up of interconnected neurons which, following an input signal represented by explanatory variables, produce an output signal, which can only be the variable being modelled. The choice of input variables is generally based on current knowledge of the processes involved. However, when the relationships between the different variables are not well known, a stepwise approach can be used. The stepwise (constructivist) approach consists of testing each of the variables individually in a reference network and combining the variables that obtain the best result for the chosen performance criterion .

Typically, the Multilayer Perceptron consists of three types of layers. Each of the variables in the input layer is connected to each of the neurons in the hidden layer(s), which are in turn connected to the neurons in the output layer(s). The connected neurons operate in a specific way and perform a weighted sum of the variables of the previous layer, following the free parameters (weights and biases) initialized beforehand with random values. Then, using a linear or non-linear activation function, they generate a result according to a well-defined algorithm [23] .

The Back Propagation Neural Network (BPNN) algorithm is the most widely used neural network method and determines the optimal weighting of features by iteratively modifying the hidden nodes and the learning rate while calculating the relative weights of the input variables between the neurons in the different layers of the neural network. In statistics, BPNNs are techniques known as classical gradient-based error correction algorithms. This principle forms the basis of gradient algorithm methods, which are effectively used in Multilayer Neural Networks (MLNs) [24] . The aim of the gradient algorithm is to converge iteratively towards an optimized configuration of weights and biases.

The weights and biases used in the model design are given by Equations (1) and (2).

$w_{ij}=w_{ij}-\alpha \left(\frac{d_{\left(MES\right)}}{d_{w_{ij}}}\right),$ (1)

$b_{j0}=b_{j0}-\alpha \left(\frac{d_{\left(MES\right)}}{d_{b_{j0}}}\right),$ (2)

where alpha (α) is the learning rate. The process of forward propagation, cost calculation, and backward propagation is repeated for a fixed number of iterations or until the cost function converges. MES is the mean squared error.

Furthermore, the optimal choice of network parameters, such as the number of layers and hidden nodes in a layer required to achieve a particular objective, is not easy to achieve. Clearly, there is no analytical tool for determining the ideal number of hidden neurons. In this situation, some authors have proposed a few rules. [25] suggest that, in the majority of applications, the optimal number of hidden neurons should be less than or equal to the number of inputs. [26] suggests that the following limit should not be exceeded, depending on the number of inputs according to Equation (3). However, these rules of thumb depend on the nature of the data used and the noise in the data. They cannot, therefore, be generalized. For the optimal choice of network parameters, it is therefore necessary to adopt a “trial-and-error” approach according to [22] .

$m\le 2\beta +1,$ (3)

where m is the number of hidden neurons and β is the number of input variables.

In general, a neural network must be capable of performing a particular function in a given environment. This capability is acquired through the learning algorithm adopted by the neural model. This is the first phase in the life cycle of a neural network (passage from a state of ignorance to a state of knowledge). In the event of a change in function or external environment, this phase must be repeated to adapt the behavior of the neural network to the new parameters (transition between two states of knowledge). Learning a neural network involves subjecting the system to a set of iterative stimuli. By intuition, the network will become better and better informed about the input set. With each learning iteration (feeding the neural network with a stimulus), the neurons in the network adapt their processing to meet the needs of the targeted function [27] .

In the majority of existing neural networks, the behavior of a neuron depends on the weights and the activation threshold associated with it. These values, known as the free parameters of the neural model, define the behavior of the neural network on all the inputs. It is therefore difficult, and even impossible, to predict the values of the free parameters during the design phase. Hence, the need to go through the learning phase. During this phase, the free parameters will be modified in order to reduce the error in the system’s response to the examples presented at its input and to experience. After a certain number of learning iterations, the margin of error decreases so that it is tolerated by the system. At this point, the neural network no longer needs to learn. Consequently, we can say that the neural network is capable of analyzing real examples from its external environment as an expert. At this point, the decision phase, often called the recall phase, begins [28] .

In our case, the neural network is trained by trial and error according to equations 5 and 6 defined above until the best intelligent model is obtained. The intelligent model obtained at the end of training was evaluated according to the performance criteria chosen, namely the coefficient of determination (R2, closer to 1) and the mean square error (MSE, smaller). These performance criteria (MSE and R2), described by Equations (4) and (6), present the best values for testing the performance of a model for optimizing the application of quantities of a chemical fertilizer.

$\text{MSE}=\frac{{{\displaystyle \sum }}_{i=1}^n{\left(N_{observed,i}-N_{simulated,i}\right)}^2}{n},$ (4)

$R=\sqrt{1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(N_{observed,i}-N_{simulated,i}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(N_{observed,i}-\bar{N}\right)}^2}},$ (5)

$R^2=1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(N_{observed,i}-N_{simulated,i}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(N_{observed,i}-\bar{N}\right)}^2},$ (6)

where $n$ is the number of measurement points, $N_{observed,i}$ is the observed value of nitrogen, $N_{simulated,i}$ is the predicted amount of nitrogen, and $\bar{N}$ is the mean value of $N_{observed,i}$.

The output value giving rise to the quantity of nitrogen supplied, depending respectively on the number of input nodes, hidden nodes, and output nodes, is given by Equation (7).

$y_k=f_0\left[\underset{i=1}{\overset{m}{{\displaystyle \sum }}}{W}_{kj}\times f_n\left(\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{W}_{ij}{x}_i+b_{j0}\right)+b_{k0}\right],$ (7)

where $y_k$ are the output values of the kth neuron at the nth iteration and $x_i$ are the input values of the network, $m$ and $n$ are the numbers of neurons in the hidden and output layers respectively; $W_{ij}$, the connection weights between the input layer and the hidden layer; $W_{jk}$, the connection weights between the hidden layer and the output layer; $b_{j0}$ and $b_{k0}$ are respectively the bias of the jth hidden neuron and the bias of the kth output neuron; $f_n$ and $f_0$ are respectively the transfer function of the hidden neuron and the output neuron.

For this purpose, the MATLAB simulation code was run to find the optimal model according to fixing criteria defined as follows: the maximum number of iterations fixed at 1000, the target error MSE and the minimum performance gradient fixed at 10⁻³ and 10⁻⁵, respectively. Levenberg-Marquardt (LM) convergence was used, and logarithmic and linear sigmoid activation functions were used, respectively, for the hidden and output layers.

Nitrogen optimization was carried out following the simulation of rice yield as a function of increasing nitrogen doses. This simulation was carried out using all the data and physico-chemical parameters of the soil and irrigation water obtained during the rainy season (April). The justification for choosing this period was inspired by the work of [29] in France and [30] in Belgium, where the fertilizer requirements of rice would be relatively much more significant for their growth. The data set consists of fine soil density (dt, g∙cm⁻³), moisture at field capacity (θcc, %), moisture at wilting point (θPF, %), organic nitrogen content of the layer studied (Norg, kg∙ha⁻¹), cation exchange capacity (CEC, Cmol∙kg⁻¹), physico-chemical soil parameters (pH, K, P, OM, TOC), rice cropping coefficient (Kc), rice yields as a function of increasing nitrogen doses (Rd, kg∙ha⁻¹), irrigation water parameters ( ${\text{SO}}_4^{2-}$, Mg²⁺, Ca²⁺, Na⁺, ${\text{NH}}_4^{+}$, mg∙L⁻¹), rice water requirement (Be, m³). The input variables were selected using the focused Principal Component Analysis (fPCA) method under R 3.1.3 software. A correlation analysis was carried out in order to identify the variables in the dataset showing significant correlations with rice yields (Rd, kg∙ha⁻¹) for a selected significance threshold of p < 0.05. Thus, compared with the work of [22] , parameters that were highly correlated (1 ≥ |r| > 0.5) with yield (Rd) were retained as explanatory variables for the construction of the model.

Finally, in order to reduce the complexity of the network and accelerate its convergence, pre-processing was carried out, mainly concerning the normalization of input and target data, based on the Min-Max method. This involves transforming the values of the input parameters and the values of the data into the range 0 to 1 according to Equation (8), given by

$\bar{X}=\frac{X-X_{\min }}{X_{\max }-X_{\min }},$ (8)

where $\bar{X}$: normalized value, X: raw analyzed value, X_min: minimum value, and X_max: maximum value.

Table 3 presents the physico-chemical characteristics of irrigation water from the N’Zi river for the July 2021 season at M’Bahiakro. Analysis of this table shows that the irrigation water is acidic and has an average temperature between 25˚C and 26˚C, with an average acidic pH value of around 6.82. Electrical conductivity (EC) remains below irrigation water standards (<3000 µs/cm). Apart from ammonium ( ${\text{NH}}_4^{+}$), the irrigation water complies with irrigation standards for nitrite ( ${\text{NO}}_2^{-}$ < 3 mg/L) and nitrate ( ${\text{NO}}_3^{-}$ < 30 mg/L).

The results of analyses carried out on soil samples from the irrigated perimeter are shown in Table 4 . The pH values obtained indicate slightly acidic soils with average levels ranging from 5.42 to 7.02, within the optimum growth range for rice species (5.5 ≤ pH ≤ 7.5). The average levels of total organic carbon (TOC) and assimilable phosphorus (P) were higher than the reference standards, with average values of 2.71% and 532.3 ppm, respectively. The average levels of organic matter (OM) (4.63%), cation exchange capacity (CEC) (12.93 Cmol/kg), and potassium (K) (1.17 Cmol/kg) are within the normative range. Furthermore, in all the soil samples studied, the average nitrogen content (0.07%) is generally low compared with the reference range, which is between 0.2% and 0.25%. Nitrogen supplementation would therefore be essential for improving rice production in the M’Bahiakro irrigated perimeter.

The results of the physical soil parameters used in this study are summarized in Table 5 . At the perimeter, the density of fine soil (Dt) between 0 and 30 cm depth averaged 70%. Moisture at field capacity and moisture at the point of permanent wilting were 17.57% and 30.6%, respectively. The crop coefficient for the irrigated perimeter averaged 0.8.

The yields obtained from rice production experiments at M’Bahiakro are summarized in Table 6 below. The application of nitrogen doses to irrigated rice crops in M’Bahiakro indicates an average yield of around 4406 kg∙ha⁻¹ for increasing doses of nitrogen varying between 70 kg∙ha⁻¹ and 140 kg∙ha⁻¹. The yield responses of rice following application of doses of 70 kg∙ha⁻¹, 90 kg∙ha⁻¹, and 140 kg∙ha⁻¹ were 4150 kg∙ha⁻¹, 4500 kg∙ha⁻¹, and 4570 kg∙ha⁻¹, respectively.

The choice of input variables for the simulation model of the response of rice to increasing doses of nitrogen was made using the fPCA method ( Figure 4 ). This method was used to highlight the relationships between rice yields (Rd) and the physico-chemical parameters of the water and soil studied. Figure 4 thus shows that Rd is significantly negatively associated with K, P, Norg, OM, θcc (−1 > r ≥ −0.8) and positively with CEC, Kc (1 > r ≥ 0.8). These significant correlations reveal the contribution of these different physico-chemical parameters (K, P, Norg, OM, θpf, CEC and Kc) to rice production yield at M’Bahiakro. These parameters are therefore used to design the simulation model.

Yellow dots indicate negative correlation values with the dependent variable in the center; green dots indicate positive correlation values with the dependent variable in the center. The red circle indicates the significance threshold (p < 0.05); the points inside indicate significant correlations with the score of interest located at the center of the focused PCA.

The BPNN neural model specified by the constructive approach, with seven (7) input variables, one (1) hidden layer of six (6) neurons, and rice yield as the only (1) output variable, is illustrated in Figure 5 . Figure 5 defines the architecture of the BPNN model for the season under study. This architecture also shows the weights (wij) generated on the network arrows and the bias (bj) corresponding to variable i and neuron j in the model ( Table 7 ). Minimum values of these free parameters (weights, bias) tend to make the models more coherent and efficient. The weights of these different parameters towards the hidden layer vary between −2.15 and 3.20. From the hidden layer to the output layer, they vary between −1.78 and 2.27. The table also highlights the biases at the level of the six neurons in the hidden layer, which are between −2.77 and 2.72, and the bias at the level of the neuron in the output layer, which is around 0.753. The results of these free parameters were used to simulate rice yields.

The indices relating to the R and MSE values obtained during the learning phase of the BPNN model ( Figure 6 ) indicate the performance of the proposed model. Analysis of the results obtained shows that, for the season studied, the R coefficients are relatively high and greater than 0.90. In addition, the MSE values obtained for this BPNN model are clearly satisfactory (MSE around 0.001 fixed). According to the performance criteria, the BPNN model developed with the input variables Norg, P, OM, K, CEC, θpf, and Kc performs well with R and MSE values ranging from 90% to 100% and from 0.48323 to 0.00174, respectively.

Table 8 shows the observed and simulated rice yields for the BPNN model developed. The simulated yields are obtained by varying the nitrogen doses in the model while setting the values of the other input parameters and the free parameters of the model following the learning process. These yields represent the response of the rice to increasing doses of nitrogen.

The difference between the observed and simulated yields is relatively small in relation to the MSE error values close to zero. In general, 80% of the MSE values obtained are satisfactory, with values relatively lower than 0.5. The architecture of the BPNN model developed would therefore confirm the performance of the 7-6-1 configuration neural network in calculating rice yields, with good superposition of the values observed experimentally and those simulated overall, with the exception of the yield obtained for 0 kg∙ha⁻¹ of Nitrogen (MSE = 0.687 > 0.5). Furthermore, the association between observed and simulated rice yields given by the linear regression line ( Figure 7 ) indicates the reliability of the model developed, with a high coefficient of determination R2 of the order of 98%.

Error ≤ 0.05 (Excellent), 0.05 < Error ≤ 0.5 (Good), Error > 0.5 (Poor).

The quadratic regression from the BPNN model illustrates the response of rice to increasing doses of nitrogen ( Figure 8 ). An increase in yield is observed, followed by a stabilization of yield with increasing doses of nitrogen. In the absence of nitrogen, the rice yield is estimated at 454 kg∙ha⁻¹. This production increases with an application of 70 kg∙ha⁻¹ of nitrogen and would reach a yield of around 3989 kg∙ha⁻¹. Beyond the dose of 70 kg∙N∙ha⁻¹ of nitrogen, production would stabilize at around 4500 kg∙ha⁻¹ in terms of yield, whatever the dose of nitrogen applied. Furthermore, the derivative of the equation y = −0.3364x² + 79.405x (y' = −0.6728x + 79.405) cancels out at a value of 118. The biophysical recommendation of nitrogen would then be capped at 118 kg∙N∙ha⁻¹ nitrogen in the irrigated plots of M’Bahiakro.