In this study, the irrigation water infiltration rate (IR) is defined by input variables in linguistic terms using a fuzzy-logic approach. A fuzzy-logic model was developed using data collected from published data. The model was trained with three fuzzy membership functions: triangular (‘trimf’), trapezoid (‘trapmf’), and pi (‘pimf’). The fuzzy system considered the number of irrigation events, applied water depth, polyacrylamide application rate, water application time, water electrical conductivity, soil surface slope, and soil texture components as input variables. The inputs were classified in terms of low, medium, and high levels. The output variable (

_{sx}(skewness coefficient)

_{w}(water applied depth)

_{c}(water electrical conductivity)

_{x}(kurtosis coefficient)

^{2}(coefficient of determination)

_{d}(standard deviation)

_{w}(water application time)

_{max}(maximum value)

_{mean}(mean value

_{min}(minimum value)

_{o,i}(observed value)

_{o,max}(maximum observed value)

_{o,min}(minimum observed value)

_{p,i}(predicted value)

_{o}(averaged observed values)

_{p}(averaged predicted values)

The infiltration of irrigation water has an important role in the process of water management and the effects of soil ponding on the uniformity of irrigation distribution. There are many economic possibilities for improving water infiltration, including the application of polyacrylamide (PAM) (

Whereas various types of PAM have been used since the 1950s, its expansion was not seen until the last decade (

The application timing and type of PAM may correspondingly affect water infiltration. It is likely that any reduction in soil infiltration is a result of the timing of the application and/or of an incorrect dosage of PAM.

With the rapidly evolving technologies in the field of irrigation measurement, it is desirable to merge the experience of many irrigation schemes with algorithms that may aid in difficult forecasting situations. Fuzzy logic is one such method that has been used in an emerging set of problem-solving algorithms.

In recent years, many researchers have used artificial intelligence techniques such as fuzzy logic. It is proposed and elaborated by

Fuzzy logic is a more flexible and intuitive approach that uses simple mathematical concepts and is tolerant to inexact data. A fuzzy system can be created to match any set of input–output data.

Fuzzy-logic system applications have been used in estimating the daily reference evapotranspiration with fewer parameters for irrigation scheduling (

Considering the cost of human operators and the instability of human behavior, an automatic approach can be a preferred alternative for controlling a high-efficiency irrigation system. Therefore, the objective of the present research is to develop a fuzzy-logic model to predict the irrigation water infiltration rate (IR) with PAM under sprinkler irrigation to improve on-farm irrigation efficiency.

The collected data were obtained from published experiments by

The published experiments were conducted in a laboratory using five soil textures with relatively high proportions of silt and sand (
^{3}
according to the soil texture type. The soil water content was measured and was found to be nearly constant before each run based on soil type. Spray nozzles, fitted with regulator valves, were installed on a pipeline at a convenient height above the surface soil box to prevent water loss. This type of nozzle is used on low-pressure center-pivot sprinkler irrigation systems. The pipeline was connected to the water supply tank. Various levels of water electrical conductivity (0.41–1.9 dS/m) were used in these experiments. Water was pumped using an electrical pump to provide different applied water depths for various applied times (

PAM, a water-soluble high-molecular-weight anionic polymer, was prepared in solutions of approximately 0–25.5 µg/mL (

The metal soil boxes had collectors at the down-slope side to provide runoff to convey water into each container at different times for estimating the infiltrated volume as the difference between the applied water volume and the measured runoff volume (

With no application of PAM, the data reflected the effect of the sequence of irrigation events using different water qualities on the infiltration rate for various soil types with varying surface slopes (

The application of PAM improved the infiltration properties of all the soils in the first irrigation. In the presence of sufficient sand particles (sandy loam soil) and a high application rate of PAM (6 kg/ha), the infiltration increased significantly with different water qualities, especially in the first irrigation. However, in silt clay loam soil, the relatively high percentage of clay and silt particles corresponded to a negligible response to PAM due to soil pore blockage and crusting, which were due, in turn, to the impact of water droplets that enhanced the clay dispersion and movement (

The fuzzy-logic system was formulated in the fuzzy-logic toolbox of MATLAB software using the Mamdani minimum–maximum inference engine. The main idea behind the Mamdani engine is to describe the process states by linguistic variables, which are defined as variables (the values of which are sentences), and to use these variables as inputs to the control rules.

To build a fuzzy system, fuzzy sets are derived solely from experience. The flowchart of the fuzzy-logic for modeling the water infiltration rate is represented schematically in
_{w}
), PAM application rate, water application time (T
_{w}
), water electrical conductivity (E
_{c}
), and soil surface slope (S)) are transformed into fuzzy variables. These variables are described in linguistics terms (low (L), medium (M), and high (H)) to address all possible fuzzy inputs (
_{mean}
), minimum value (
_{max}
), standard deviation (
_{d}
_{x}
_{sx}

The developed fuzzy-logic model relies on 55 rules when using the ‘trimf’ and 53 rules for both ‘trapmf’ and ‘pimf’. The number of rules represents all possible combinations of the categorized system inputs. The product of these fuzzy sets forms a fuzzy patch, which is an area that represents the set of all associations that the rule forms between those inputs and outputs. The fuzzy rules define a set of overlapping patches that relate a full range of inputs to a full range of outputs. All uncertainties or nonlinear relationships are included in the descriptive fuzzy inference procedure in the form of IF–THEN statements.

Fuzzy logic is based on rules of the form IF–THEN that convert inputs to outputs. The IF portion of a rule refers to the degree of membership in one of the fuzzy sets. The THEN portion refers to the consequence or the associated system’s output fuzzy set. For fuzzy inference, the Mamdani system was used, which is considered an AND method that is used as a min (minimum) activation operator (

The coefficient of determination (
^{2}
) expressed below measures the degree of correlation among the observed and predicted values (from the fuzzy-logic model) of the variable IR, with values close to 1.0 indicating a good model performance:

where
_{o,i}
_{p,i}
_{o}
_{p}

The root-mean-squared error (RMSE) was used as a criterion to judge the accuracy and reliability of the model. The RMSE has the advantage of expressing the error in the same units as those of the variable, thus providing more information regarding the efficiency of the model (

The mean absolute error (MAE) measures the average magnitude of the errors in a set of forecasts without considering their direction. The MAE is given by

The RMSE and MAE can be used together to diagnose the variation in the errors in a set of forecasts. The RMSE will always be greater than or equal to the MAE; the greater the difference between them, the greater is the variance in the individual errors in the set. If the RMSE equals the MAE, then all of the errors are of the same magnitude. The values of RMSE and MAE can range from zero to ∞, and lower values indicate a more accurate model.

A model efficiency (ME) value of 1.0 indicates a perfect fit between the measured and predicted data. This value can be negative and was calculated using the following equation (

The overall index of model performance (OI) combines the normalized root-mean-square error (which is the RMSE divided by the range of observed values) and the model efficiency indicators to verify the performance of the mathematical models. An OI value of one for a model indicates a perfect fit between the observed and predicted data (

where
_{o,max}
_{o,min}

The steadiness of the developed fuzzy-logic model of the IR was verified by comparing to the results obtained by experimental measurement.

As can be observed, an acceptable agreement is illustrated in
^{2}
value was 88.2% for the IR values modeled from ‘trimf’ and approximately 93.1% for the other two functions.

The above statistical parameters indicate that the ‘trapmf’ and ‘pimf’ performed the best, with an
^{2}
value that was approximately 5.5% better than that from the ‘trimf’. The RMSE value for the fuzzy-logic model using ‘trimf’ was 22.5 and 28.2% less accurate than those using the ‘trapmf’ and ‘pimf’, respectively. For the ME and OI calculations, the ‘trapmf’ was 7.4 and 4.8% more accurate, respectively, than that of ‘trimf’. While the ‘pimf’ was 8.7 and 5.7% more accurate, respectively.

The aforementioned results illustrate that the values of IR evaluated by the fuzzy-logic model using the ‘pimf’ were more exact than those using the ‘trapmf’. Thus, the selection of membership functions in terms of shape and boundary had an obvious effect on the determination of the IR values. Therefore, the fuzzy-logic model is considered acceptable for the prediction of IR values and may lead to significant improvements in field water management by its inclusion in automated irrigation systems.

There are nine input variables (R, D
_{w}
, PAM, T
_{w}
, E
_{C}
, S, Cl, Si, and Sa) and one output variable (IR) using the ‘trimf’ (
_{w}
, S, and Si values of 2, 15 min, 7.5%, and 49.1% were medium, low, high, and high sets, respectively, with degrees of membership (µ) of one. E
_{C}
and Cl, which were equal to 0.5 dS/m and 23.2%, were low and medium sets with degrees of membership of 0.85 and 0.93, respectively. The remaining antecedents are D
_{w}
and PAM, which were equal to 23.9 mm and 4 kg/ha and were members of the medium (µ = 0.57 and 0.58) and high (µ = 0.18 and 0.17) sets, respectively. Similarly, a value of Sa of 27.7 is part of the low and medium sets with different degrees of membership of 0.5 and 0.25, respectively. The fuzzy rules assume the following structure:

Rule 12: IF (R is M) and (D
_{w}
is not L) and (T
_{w}
is L) and (E
_{C}
is L) and (PAM is not L) and (S is H) and (Cl is M) and (Si is H) and (Sa is not H) THEN (IR is M).

Rule 13: IF (R is M) and (D
_{w}
is not L) and (T
_{w}
is L) and (E
_{C}
is L) and (PAM is not L) and (S is H) and (Cl is M) and (Si is H) and (Sa is not H) THEN (IR is H).

In the second step,
_{C}
). The rule strength is then 0.85. Similarly, the second rule strength is 0.85.

The third step is the combination of the working rules (aggregation method) that were produced from the implication method. If the several sets of rules above share the same consequence or output, the value of the output is assigned by the value of the strongest rule. The limiting values for both rules associated with a medium and high probability of IR are then assigned the value 0.85. These values were used for each output fuzzy set.

Finally, a value of IR of 23 mm/h was obtained after the use of the COG defuzzification method (

where Y is a crisp value of the output and

As conclusions, the use of a fuzzy-logic system as a decision-making technique was introduced for predicting IR under a sprinkler system. The fuzzy-logic model was developed by employing triangular, trapezoid, and pi membership functions (‘trimf’, ‘trapmf’, and ‘pimf’) for the input and output variables. The nine input variables included the number of irrigation events (R), applied water depth (D
_{w}
), polyacrylamide application rate (PAM), water application time (T
_{w}
), water electrical conductivity (E
_{C}
), soil surface slope (S), and percentage of clay (Cl), silt (Si), and sand (Sa) particles in the soil texture. The statistical criteria indicated that the compatibility between the experimental and computed data was acceptable, which confirmed that simulations are capable of successfully reproducing the values of IR using a fuzzy-logic model when using the ‘trapmf’ and ‘pimf’ only. The ‘pimf’ produced the best results. Thus, the developed fuzzy-logic model can effectively estimate IR values using the previous input variables. This model is attractive for use as part of an intelligent irrigation management system.