Introduction

One of the most important factors in irrigation and drainage networks is drain spacing. The agricultural land suitability for kinds of irrigation and drainage networks is defined as the process of land performance assessment with determination of soil properties (Diallo et al. 2016; Ahmed 2016). Determination of saturated hydraulic conductivity (ks) is one of the most difficult factors in drain spacing measurement (Van Schilfgaarde 1970; Bouwer and Jackson 1974; Topp et al. 1980; Puckett et al. 1985; Dorsey and Bair 1990; Gupta et al. 1993; Mohanty et al. 1994). In classic estimation; like Thiessen method to estimate variables, the location of data was not considered at the area and data coefficient was not dependent for estimation but in geostatistical view both factors are considered, in these methods, error and also sampling error rate is measured and, therefore, an index is provided for space structure strength and data's local relation. Due to these reasons, engineers are encouraged to use geostatistical tools in combination with GIS (Hoseini and Kamrani 2018) for integrating and handling multiple and iindicators dependent on the coordinates of points (Albaji et al. 2015; Tuik 2018). Those techniques provide structured and spatially explicit evaluation frameworks (Karimi et al. 2018) and facilitate evidence-based judgments for sustainable land-use management practices (Roy et al., 2018). Begay Herchgani and Heshmati (2012) examined Shahrekord's groundwater quality by using geo-statistical for use in irrigation systems. In this study, Electrical conductivity (EC), Total dissolved salts (TDS), Total suspended solids (TSS) and water pH in the 97 wells have been measured. Also the Ordinary Cokriging method for variables zoning was used and also to demonstrate the accuracy of the estimations root-mean-square error (RMSE) criterion was used. In the all indicators less than 40% were obtained that indicates the estimates accuracy is good. The obtained results represent the views solidarity matching and calculated map that showed good accuracy of Variogram models and the Ordinary Cokriging estimator in the interpolation and qualitative indicators zoning of Shahrekord’s groundwater. Hoseini et al. (2017) compared genetic algorithm (GA) and geostatistical analysis with Neural Network method to estimate soil saturated hydraulic conductivity with properties of particle size distribution. The soil data were gathered of 134 soil profiles points. Results showed that Ordinary cokriging has the best fit for the geostatistical methods. And between these methods, the geostatistical method has the best correlation with the soil properties. Nezami and Amirpour (2012) provided soil salinity map by using geostatistical methods on the Qazvin plain of Iran. Because on the ground assessments, soil salinity is a limiting factor for growing all of plants, Researchers took sample from 0 to 30 cm of soil depth on the Qazvin plain and measured electrical conductivity (EC), pH. In this study, several methods including cokriging, Spline and the inverse distance weighting method were used to assess soil salinity. Results showed that Auxiliary tools increase the accuracy of the kriging interpolation method. In this study, the percentage of clay was used as an aid in the cokriging method and the results showed that the accuracy can be increased with this method. Naseri et al. (2009) used the kriging method to check soil quality for choosing different irrigation methods in southern Iran. In this study, six factors including soil texture, soil depth, lime, salinity, drainage and slope were considered. The results showed that 1732 hectares (48.5%) of the study area are suitable for all three types of irrigation methods (sprinkler, surface and drip). Geostatistical methods can be used to minimize the error of evaluation soil indicators to evaluation a low cost of hydraulic conductivity. Therefore, the purpose of this study is to compare the geostatistical methods of Kriging and conventional Thiessen method to determine the hydraulic conductivity of the soil and its expansion at the regional area, in order to increase the accuracy of measuring the hydraulic conductivity in different drainage projects.

Materials and methods

The area of the study

Vaise irrigation and drainage network are a part of land mugs of lower Karkheh area in Khuzestan province. Lands of Vaise area in the right side of Karkheh River are located 32 km far from west of Ahwaz city and 7 km far from west of Hamidieh city. It lies within 31°29' to 31°35' longitude and 48°16' to 48°23' latitude. Gross and net surface of these lands are 3200 and 2719 hectares, respectively. Vaise region in the north and northeastern is surrounded by Allah Akbar hills and the East River near the Karkheh is located. Studied areas have long hot summers with low humidity and mild and short winters. Minimum and maximum temperatures are related to July and January, respectively. Annual rainfall in these areas is low and its distribution is more from February to December. General plan of Vaise irrigation network and sampling points is shown in the Fig. 1.

Fig. 1
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Spots of hydraulic conductivity measuring points

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Evaluation parameters

Cross-validation helps in deciding the model which makes the best predictions. The calculated statistics serve as diagnostics that indicate whether decisions on choice of method of kriging, as well as on choice of transformation and order of trend decided, are reasonable to provide an unbiased and more accurate prediction of parameter values along with valid prediction standard errors. Cross-validation tool of geostatistical wizard gives a scatter plot of predicted vs measured values and a best fit linear line through it. Higher the slope of this line (magnitude always < 1), nearer it will be to 1:1 line which is an indicator of good prediction choices. If all the data are spatially independent, best fit line would be horizontal. For a model that provides unbiased predictions, the mean of prediction errors should be close to zero. Again for the correct assessment of the variability and to check if the prediction standard errors are appropriate and valid, the root-mean-square prediction error and average standard prediction error should be similar and the root-mean square standardized prediction error should be close to 1. (Kalpana and Aggrawal 2011).

Several researches have shown that each geostatistical estimation method for certain variables, is appropriate so to investigate the spatial variability and interpolation of each feature, several geostatistical methods should be compared and evaluated and choose among them, the best way depending on the desired accuracy and ease of use. In this, comparing technique was mutual authentication method. This technique is based on that each time an observation point was removed and for it from the adjacent points, some value is estimated, so the actual value is returned to the previous location and for all grid points, this operation is repeated. Finally, given the observed and estimated values, to assess the performance of the model, the maximum error (ME), root mean square error (RMSE), relative percentage error (ε), mean absolute error (MAE), Goodness of fit (R2), coefficient of residual mass (CRM) and model efficiency (EF) were used. The best predictions occur when ME, RMSE, CRM, ε, and MAE statistics tend to zero and EF and R2 tend to one. The Eqs. 1–6 show the equations of the mentioned statistics.

$$ {\text{RMSE}} = \sqrt {\frac{1}{N}\sum\nolimits_{i = 1}^{N} {\left( {Q_{i} - P_{i} } \right)^{2} } } $$
(1)
$$ \varepsilon = \frac{100\% }{N}\mathop \sum \limits_{i = 1}^{N} \left| {\frac{{Q_{i} - P_{i} }}{{Q_{i} }}} \right| $$
(2)
$$ {\text{MAE}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left| {Q_{i} - P_{i} } \right| $$
(3)
$$ R^{2} = \frac{{\left[ {\mathop \sum \nolimits_{i = 1}^{N} \left( {P_{i} - \overline{P}} \right)\left( {Q_{i} - \overline{Q}} \right)} \right]^{2} }}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {P_{i} - \overline{P}} \right)^{2} \mathop \sum \nolimits_{i = 1}^{N} \left( {Q_{i} - \overline{Q}} \right)^{2} }} $$
(4)
$$ {\text{EF}} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Q_{i} - \overline{Q} } \right)^{2} - \mathop \sum \nolimits_{i = 1}^{N} \left( {P_{i} - Q_{i} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Q_{i} - \overline{Q} } \right)^{2} }} $$
(5)
$$ {\text{ CRM}} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} Q_{i} - \mathop \sum \nolimits_{i = 1}^{N} P_{i} }}{{\mathop \sum \nolimits_{i = 1}^{N} Q_{i} }} $$
(6)

In the above relations, N: sample number, Pi: predicted values by the model, Qi: real values, \( \overline{p}\): mean values predicted by the model, \( \overline{Q}\):the mean of real values.

Results and discussion

In draining projects, hydraulic conductivity is used for area distinction, different areas were divided into three groups of [K < 0.5, 0.5 < K < 1, K > 1] (m/day). Therefore, GIS software was used to categorize Polygon Thiessen to divide areas into three areas which are shown in Fig. 2 and their areas are given in Table 1. Since interpolation kriging method, for data that their distribution is close to normal distribution provides more accurate estimates. So to determine the distribution of hydraulic conductivity data, its distribution range was determined. Since the normal distribution has a skewness of 0 and kurtosis 3, the hydraulic conductivity data due to having the skewness of 1.45 and kurtosis 2.02 doesn’t have the normal distribution. Therefore, for using the kriging method this data should convert by non-normal distribution conversion method to normal. One of these conversions is logarithmic converter. In the data with log-normal distribution, the data underwent logarithmic transfer function close to a normal distribution. Figure 3 shows the hydraulic conductivity values that are converted.

Fig. 2
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Project whole area hydraulic conductivity division

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Table 1 Average hydraulic conductivity with Thiessen method
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Fig. 3
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Transformed hydraulic conductivity frequency data

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By giving UTM coordinates to GIS software, the trend was analyzed on the data, then as shown in Fig. 4, data have trend in X and Y direction, since Kiriging method was used we should consider this issue and eliminate trend from data.

Fig. 4
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Trend in X and Y axes direction

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By plotting the directional variogram with assuming the region as non-uniform in the directions 0° and 45° and 90° also in 135° it can be seen that these variograms in the different directions are homogeneous and of the same type and the effecting range and sill effects are the same and follow homogeneity of the spatial structure of the model, on These spherical variograms model, the Nugget effect is 0.003 and the maximum sill 0.027. These directional variograms are shown in Fig. 5 and surface of direction variograms for all region was shown in Fig. 6. Since C0/Q2 < 0.5 it was proved that the obtained variogram has acceptable strength. Because models had similar parameters like Sill and Nugget effect, it was proved that the area random distribution of variable data is similar, so to obtain actual structure for whole area we drew whole area variogram in which the obtained space structure was spherical with 0.027 maximum Sill and Nugget effect of 0.003. In this method, the amount of C0/Q2 was below 0.5 and According to this structure, we estimate hydraulic conductivity in whole area that shown in Fig. 7. To evaluate whether or not bias of the obtained estimates, cross-validation method was used as can be seen in Fig. 8. The estimated value for kriging was very close to the measured values, which led the fitted line between the actual and estimated values to be close to the angle of 45°. Results show that Simple Kriging method had more acceptable for estimating hydraulic conductivity. Table 2 exhibits area of these fields and average of hydraulic conductivity. Accuracy of this estimate was analyzed by cross validation method. According to R2 = 0.89, the mentioned estimate showed desirable accuracy. Comparing Tables 1 and 2 indicated that middle hydraulic conductivity (0.5 < K < 1) and also high hydraulic conductivity (K > 1) in Kiriging calculation, comparing to Thiessen method were increased. So in calculation of distances between drains more areas were allocated to high hydraulic conductivity which led to lower actual rate between drains distance by Thiessen method and due to soil incapability to discharge more water because of actual lower hydraulic conductivity, after a while underground water comes up and draining problem occurs in these areas. However, Hoseini et al (2017) studied the spatial variability in hydraulic conductivity and observed the existence of spatial dependence in their study. The results showed that the spatial correlations between the values of hydraulic conductivity were low and the obtained variogram was less dependent on the auxiliary variables. Even when using the auxiliary variable, the nugget effect was reduced to zero. Therefore, it can be concluded that the use of auxiliary variables that directly affect the main variables. It can determine the spatial structure of a variable more accurately at the area level. Some studies have shown the relationship between hydraulic conductivity and soil properties dependence (Topp et al. 1980., Hoseini 2015), it seems that, in order to accurately estimate of saturated soil hydraulic conductivity parameter, soil texture properties can also be used soil, applying the use of kriging method, without the use of auxiliary variables In the hydraulic conductivity estimation, It may reduce the accuracy of the estimate (Basaran et al. 2011; Kalpana and Aggrawal 2011); but, considering soil texture as the effective parameter in the hydraulic conductivity, estimation is very close to the true value. One of the advantages of geostatistics over other estimation methods such as Thiessen method, is that hydraulic conductivity estimation can be noted in all areas in which measurements are done. In fact, using this method, hydraulic conductivity map is provided at the regional level that can be directly used on irrigation and drainage networks with high confidence. Hosseini (2015) investigated the spatial variability of soil hydraulic conductivity in the Ardabil province of Iran and shown that, in this area, soil hydraulic conductivity had moderate spatial correlation by comparing the Nugget effect to the threshold is over 60%. However, in this study, the amount of Nugget effect was estimated to be small. The relationship between hydraulic conductivity and soil parameters has also been studied by Castellini et al (2019), which is consistent with the results of this study. In his research, by establishing a proper relationship between soil properties and hydraulic conductivity, the amount of hydraulic conductivity was estimated with appropriate accuracy at the field area. In the Table 3, the quantitative evaluation indicators of kriging based parameters of root mean square error (RMSE), percentage relative error (ε), mean absolute error (MAE) and the coefficient of determination (R2) and other statistical indices is given. Comparison of the model show that the simple kriging model has good capability and high accuracy in predicting the saturated hydraulic conductivity of the soil, so that the kriging model is much more closely to the real data in comparison with the Thiessen model.

Fig. 5
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Estimate of hydraulic conductivity in the area by ordinary Kriging method

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Fig. 6
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Directional variogram for region

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Fig. 7
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Estimate of hydraulic conductivity in the area by simple Kriging method

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Fig. 8
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Cross validation method to analyze simple Kriging method

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Table 2 Average hydraulic conductivity for area with kiriging method
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Table 3 Performance indicators (RMSE, MAE, ε, R2, EF, CRM, ME) for models
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Conclusion

In this research, as the results showed, geostatistical analyst has acted as a bridge between geostatistical methods and geographic information system. Few geostatistical methods have been available in other existing software related to kriging analysis, but only in the geostatistical analyzer available in the geographic information system, various geostatistical methods are available and they can be used to model different soil indicators. This has made it possible to quantify the quality of prediction by measuring the statistical error of the predicted indices. Hydraulic conductivity is one of the hydrodynamic characteristics of the soil, which plays an important role in the movement and transfer of water and solutes in the soil. In drainage projects, it is necessary to determine the exact hydraulic conductivity of soil at the fields. In this study, saturated hydraulic conductivity of soil in the Vaice region using kriging and Thiessen was estimated. The results showed that simple kiriging has high accuracy in the determining the amount of saturated hydraulic conductivity of the soil. However, also, kriging has higher accuracy than the Thiessen method and simple kriging can well simulate saturated hydraulic conductivity of the soil with high accuracy and less error.