Introduction

Water management is essential for improving the efficacy and sustainability of Irrigation and agricultural systems particularly where water is limited (Siltecho et al. 2014). Water management is associated with estimation and control of water. Consequentially, these processes are dependent on the soil hydraulic conductivity which needs to be calculated in the field or in the laboratory. On the other hand, direct observation of hydraulic conductivity of soil is complex, tedious, relatively costly, labor intensive and time-consuming (Lakzian et al. 2010; Emami et al. 2012; Kalkhajeh et al. 2012; Fereshte 2014; Sihag et al. 2017a; Singh et al. 2017). Hydraulic conductivity is significant parameters for estimating infiltration rates and more hydrological process (Gülser and Candemir 2008; Angelaki et al. 2013). Thus, indirect methods using predictive approaches have been generated for estimation of hydraulic properties of soil from simply assessable soil properties (Schuh and Bauder 1986; Emami et al. 2012; Sihag et al. 2017b) However, predictive approaches of the soil hydraulic conductivity have obtain consideration by researchers and attempts have been made to raise the accuracy of predictability (Skaggs et al. 2001; Al-Sulaiman and Aboukarima 2015). The predictive approaches of the soil hydraulic conductivity are developed by different methods like multilinear regression and artificial neural network. These methods have been recommended for predicting hydraulic conductivity as encourage results of different research papers (Tamari et al. 1996; Ghanbarian-Alavijeh et al. 2010; Sarmadian and Mehrjardi 2010; Moosavi and Sepaskhah 2012; Nosrati et al. 2012; Sihag et al. 2017c) have reported.

Some recent studies recommended successful application of soft computing techniques viz. SVM, ANN, GEP, M5 tree, fuzzy logic, GRNN, ANFIS and random forest regression in the field of hydrology and water resources (Erzin et al. 2009; Aggarwal et al. 2013; Parsaie et al. 2015; Parsaie and Haghiabi 2015a, b; Tiwari et al. 2017, 2018; Sihag et al. 2018a). Keeping in the view the potential of Fuzzy logic and ANN techniques, this study was conducted to observe the efficacy of FL and ANN techniques in developing a prediction model for estimating unsaturated soil hydraulic conductivity. However, FL has shown capablity in modeling of nonlinear functions. It reads quality of the data set and arranges the system features according to the particular error criteria.

Artificial neural network (ANN)

It is a machine learning technique which is used for numerical forecast (Schaap and Leij 1998; Erzin et al. 2009; Heddam 2016; Parsaie 2016a, b; Parsaie et al. 2016; Sihag et al. 2018b). It is motivated by the performance of the nervous system and brain architecture. ANN has single input, one or multiple hidden and single output layers. Every layer consists of the quantity of nodes and the weighted connection between these layers represents the link between the nodes. Input layer having nodes corresponding to the number of input parameters, distributes the data presented to the network and does not help in processing. This layer follows one or more hidden layers which help in the processing of data. The output layer is the final processing unit. When an input layer is subjected to an input value which passes through the interconnections between the nodes, the corresponding weights are multiplied to these values and added up to achieve the total output (S j ) to the unit

$${S_j}=\mathop \sum \limits_{i} {X_{ij}} \times {a_i}$$
(1)

where X ij is the weight of interconnection from unit i to j, a i is the input value at input layer, S j is output gotten by activation function to generate an output for unit j. The detailed discussion about ANN is provided Haykin (1999). In present analysis, back propagation algorithm is used based on ANN.

Fuzzy logic (FL)

FL is an effective and efficient method to implement the nonlinear mapping between input and output variables. FL provides a simple and a ordinary way of solving the problems in which there is not clearly defined criterian but only the random variables. Fuzzy set theory solve the problems with the help of linguistically information and linguistic labels stimulated by MFs. Although the idea of FL was introduced by Zadeh (1965).

There are mainly two types of rule base, i.e. (1) Sugeno and (2) Mamdani. In this study, Mamdani type of rule and prod process based fuzzy model is used because of estimating more accurate results. Weighted average process has been applied for the defuzzification in the model of FL. The main characteristic of FL is allowed a object to partial links with various subsets, made possible with MFs. Partial belonging to a set is defined numerically by a MF, whose range is zero to one.

Choice of membership function

Membership function may be of many shapes for example triangular, generalized bell shaped, Gaussian functions and many more. In present study, performances of triangular and Gaussian shape membership functions have been compared. Both membership functions (MFs) are defined below:

$$\begin{aligned} {\text{Triangular}}\;({\text{trimf}}):{\mu _X}(a) & =({\text{a}} - {\text{x}})/({\text{y}} - {\text{x}}),\quad {\text{x}} \leq a \leq {\text{y}} \\ & ={\text{ }}({\text{z}} - {\text{a}})/({\text{z}} - {\text{y}}),\quad {\text{y}} \leq a \leq z \\ & ={\text{ }}0 \\ \end{aligned}$$
(2)
$${\text{Gaussian}}\;({\text{gaussmf}}):{\mu _X}(a)={e^{\frac{{ - {{(a - m)}^2}}}{{2{\sigma ^2}}}}},$$
(3)

where a is a function of a vector and depends on the parameters σ is the standard deviation and m is the mean used in gaussmf.

Model performance evaluation criteria

To analyze the capability of various modeling methods in estimating unsaturated hydraulic conductivity of soil correlation coefficient (CC) and root mean square error (RMSE) values were calculated using the testing dataset.

$$CC=~\frac{{n\sum HF--(\sum H)(\sum F)}}{{\sqrt {n(\sum {H^2}) - {{(\sum H)}^2}} \sqrt {n(\sum {F^2}) - {{(\sum F)}^2}} }}$$
(4)
$$RMSE=\sqrt {\frac{1}{n}\mathop \sum \limits_{{i=1}}^{n} {{(H - F)}^2}}$$
(5)

where H = observed values, F = predicted values, n = number of observations

Methodology and data set

Experiments were conducted in a hydraulic laboratory located at National Institute of Technology, Kurukshetra, India. The soil selected for experimentation was mixture of rice husk ash, fly ash and sand. Sand is the major component of the mixture, whereas rice husk ash and fly ash used as impurities. So soil type is selected as sandy soil for the calculation of Ku. All the measurements had been taken on predetermined initial conditions (bulk density and moisture contents). The moisture contents of the samples were measured with the help of electric oven for 24 h at 100 °C. The samples were carefully compacted in proctor having a volume of 1000 cm3.The properties of the material used for experimentation are shown below in Table 1. The detail of the soil samples and moisture content are listed in Table 2.

Table 1 Properties of the material used for experimentation
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Table 2 Detail of soil samples and their moisture content
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Measurement of cumulative infiltration of soil in the laboratory

The cumulative infiltration was observed in the laboratory (Fig. 1) with the help of mini disk infiltrometer (Decagon Devices Inc., 2014). Two parts (water reservoir and bubble part) are consisting in mini disk infiltrometer, which are linked that a Mariotte tube to provide a steady negative water pressure head of 0.5–7 cm (equivalent to 0.05–0.7 kPa). The base of the mini disk infiltrometer contains a porous sintered steel disc having 4.5 cm diameter and 3 mm thickness. The water filled tube is placed upon the soil surface (Fig. 6) resulting in infiltrate of water into the surface soil. Infiltration is depending on the hydraulic conductivity and sorptivity of the soils. A suction head of 1–6 cm (equivalent to 0.1–0.6 kPa) was chosen in this study. At regular time intervals, the volume of the water in the reservoir part was recorded during the observation.

Fig. 1
figure 1

Mini disc Infiltrometer (Infiltrometer User’s Manual, 2014)

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Computation of unsaturated soil hydraulic conductivity (Ku)

The Zhang (1997) method is easy and works well for measuring the infiltration into unsaturated soils, when data is recorded by mini disk infiltrometer (Decagon Devices 2014). The method needs measured cumulative infiltration vs. time and adjusting the outcome in the given relation:

$$I(t)=~{A_1}t+{A_2}\sqrt t$$
(6)

where I is the cumulative infiltration (cm), t is the time (s), and A1 (cm/s) and A2 (cm s−0.5) are parameters. Such that A1 is associated to hydraulic conductivity and A2 associated to sorptivity of the soil. The soil hydraulic conductivity (K) is then calculated from:

$${\text{K}}=\frac{{{A_1}}}{B}$$
(7)

where slope of the curve of the cumulative infiltration vs. time in square root is A1 and B is concerning with the Van Genuchten (1980) variables for a specified soil type, suction rate and radius of the mini disc infiltrometer. Figure 2 displays plot of cumulative infiltration vs. time in square root for a soil. B is computed from:

Fig. 2
figure 2

Cumulative infiltration vs. time in square root for a soil

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$$B=\frac{{11.65({m^{0.1}} - 1)\exp 2.92(m - 1.9)\alpha {h_1})}}{{{{(\alpha {r_1})}^{0.91}}}}\quad {\text{for}}\;m \geq 1.9$$
$$B=\frac{{11.65({m^{0.1}} - 1)\exp 7.5(m - 1.9)\alpha {h_1})}}{{{{(\alpha {r_1})}^{0.91}}}}\quad {\text{for}}\;m \geq 1.9$$

where Van Genuchten parameters are m and α for the soils, r 1 is the disc radius and h 1 is the suction at the disc surface.

Data set

The data set selected for the FL and ANN models was collected from the experiments conducted at in hydraulic laboratory of N.I.T Kurukshetra, as summarized in Table 3. The success of model to estimate the Ku of the soil depends upon the degree of the train data set. Out of 46 data as many as (32) datasets used as training data and remaning 14 were used for test. Input variables include sand (S), rice husk ash (Rha), fly ash (Fa), suction head (Sh), bulk density (Bd) and moisture content (Mc) whereas unsaturated hydraulic conductivity (Ku)is considered as output. The range of different input parameters applied in both ANN and Fuggy logic techniques are listed in Table 3.

Table 3 Characteristics of training and test data
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Result and discussion

Results of fuzzy logic (FL)

Developing the memadani rule based fuzzy logic models are a trial and error method. For this purpose, the collected data set, given in Table 4, was used for training and testing. Choosing the training and testing data sets was based on the randomization. Designing of FL model includes defining the number and shape of membership functions. Two shapes (triangular and Gaussian) of the membership functions are selected for generating the models. In the study, the number of membership functions was added one by one to each input and output parameters and then FL model was trained and test. The triangular and Gaussian MF for different variables are shown in Fig. 3a–f and the MF for Ku of the soil is shown in Fig. 4. The if-then rule is used in this study, which is depicted in Fig. 5a, b gives the rule viewer. Figure 6 shows the surface diagrams of two inputs and output using FL model. Results of the FL model to predict the unsaturated hydraulic conductivity of soil are shown in Fig. 7. Results of the performance evaluation parameters of FL models are listed in Table 4. As shown in Table 4, Gaussian MFs based FL model works well than triangular MFs based FL model with correlation coefficient value of 0.9270 (RMSE = 7.4393 cm/h). Overall, as shown in Fig. 7, FL based models ability is suitable for predicting values of unsaturated hydraulic conductivity of soil. It is notable that in this figure the FL (triangular) is associated to the results of the triangular membership function based fuzzy logic model and FL (Gaussian) is associated to the results of the Gaussian membership function based fuzzy logic model.

Table 4 Detail of performance evaluation parameters using ANN and FL of testing data set
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Fig. 3
figure 3figure 3

a MF of sand. b MF of rice husk ash. c MF of fly ash. d MF of suction head. e MF of bulk density. f MF of moisture content

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Fig. 4
figure 4

MF of unsaturated hydraulic conductivity (Ku)

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Fig. 5
figure 5

a Rules. b Rules viewer

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Fig. 6
figure 6

Surface diagram of different input variables and the output (unsaturated hydraulic conductivity)

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Fig. 7
figure 7

Actual vs. predicted values of Ku using FL based models of testing data

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Results of ANN model

Developing the ANN model (e.g. number of neurons in hidden layer, number of hidden layers, momentum, learning rate, Iteration etc.) is a trial and error process. The ANN model consist one hidden layer. Hidden layer contain eight neurons with momentum = 0.2, learning rate = 0.1 and Iteration = 1500. The performance of the ANN model is shown in Fig. 8. As shown in Table 4, the values of CC and RMSE of ANN model are 0.8662 and 4.5907 (cm/h) for testing period. Overall, assessing Fig. 8 and Table 4 shows that performance of ANN model is suitable for predicting the unsaturated hydraulic conductivity (Ku) of soil.

Fig. 8
figure 8

Actual vs. predicted values of Ku using ANN of testing data

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Comparison of models

Comparison of soft computing models (Table 4) indicates that ANN model works well than Fuzzy logic based models. Table 4 also indicates that triangular MFs based FL model works well than triangular MFs based FL model. To compare the performance of FL and ANN models a agreement was plotted in Fig. 9 with testing data set. It can be incidental from the figure that the estimated value produced by ANN were in extremely near proximity to the actual unsaturated hydraulic conductivity. Keeping in view the improved performance of ANN modeling approaches, a graph between test data set number and Ku is plotted (Fig. 10). It can be inferred from the Figure that predicted value provided by ANN were close to the actual Ku and the predicted Ku values were found to follow the same pattern as that of actual Ku.

Fig. 9
figure 9

Actual vs. predicted values of Ku using FL and ANN based models of testing data

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Fig. 10
figure 10

Variation in values of Ku using different approaches in comparison to the actual values of Ku

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Conclusion

Present study is an attempt in direction of estimating the unsaturated hydraulic conductivity of soil (Ku) using two soft computing techniques namely by ANN and FL. It is concluded from the study that both approaches work well with this dataset. From the comparison of performance evaluation parameters, it has been observed that ANN approach works well with a coefficient of correlation = 0.8662 (RMSE 4.5907 cm/h) in comparison to FL approaches. The other conclusion is that Gaussian MF based FL approach works well than triangular MF based FL approach with correlation coefficient = 0.9270 (RMSE 7.4393 cm/h).