Geostatistical interpolation model selection based on ArcGIS and spatiotemporal variability analysis of groundwater level in piedmont plains, northwest China
 Yong Xiao^{1},
 Xiaomin Gu^{1},
 Shiyang Yin^{1, 2}Email author,
 Jingli Shao^{1},
 Yali Cui^{1},
 Qiulan Zhang^{1} and
 Yong Niu^{2, 3}
Received: 27 December 2015
Accepted: 30 March 2016
Published: 11 April 2016
Abstract
Based on the geostatistical theory and ArcGIS geostatistical module, datas of 30 groundwater level observation wells were used to estimate the decline of groundwater level in Beijing piedmont. Seven different interpolation methods (inverse distance weighted interpolation, global polynomial interpolation, local polynomial interpolation, tension spline interpolation, ordinary Kriging interpolation, simple Kriging interpolation and universal Kriging interpolation) were used for interpolating groundwater level between 2001 and 2013. Crossvalidation, absolute error and coefficient of determination (R^{2}) was applied to evaluate the accuracy of different methods. The result shows that simple Kriging method gave the best fit. The analysis of spatial and temporal variability suggest that the nugget effects from 2001 to 2013 were increasing, which means the spatial correlation weakened gradually under the influence of human activities. The spatial variability in the middle areas of the alluvial–proluvial fan is relatively higher than area in top and bottom. Since the changes of the land use, groundwater level also has a temporal variation, the average decline rate of groundwater level between 2007 and 2013 increases compared with 2001–2006. Urban development and population growth cause overexploitation of residential and industrial areas. The decline rate of the groundwater level in residential, industrial and river areas is relatively high, while the decreasing of farmland area and development of watersaving irrigation reduce the quantity of water using by agriculture and decline rate of groundwater level in agricultural area is not significant.
Keywords
Background
Scarcity of water has become an important issue worldwide (Kahil et al. 2014; Solomon 2015; Karmegam et al. 2010). As the major irrigation water source in arid and semiarid regions, groundwater is an important water source for the development of human society (Smedema and Shiati 2002). While overexploitation causes a continuous decline in groundwater level. To estimate the degree of overexploitation, the surface trend of groundwater should be known, which can be determined by the available well data integrated in various interpolating methods. Therefore, selecting the best method to estimate the temporal and spatial variations of groundwater level has an important strategic significance for reasonable groundwater management and sustainable development of water resources.
Spatial interpolation is a method to estimate the data in contiguous area and forecast the unknown points (the information is missing or cannot be obtained) with available observation data (Chai et al. 2011; Losser et al. 2014), including geostatistical interpolation and deterministic interpolation. Geostatistical interpolation consists of ordinary Kriging interpolation (OK), simple Kriging interpolation (SK) and universal Kriging interpolation (UK); deterministic interpolation comprise global polynomial interpolation (GPI), local polynomial interpolation [inverse distance weighted interpolation (IDW), planar spline interpolation and local polynomial interpolation (LPI)].
The geostatistical interpolation, or local space interpolation, is known as the unbiased optimal estimation method for describing regionalized variables (Gundogdu and Guney 2007), but has some defects in smooth effects. The local errors can be corrected without reducing the accuracy by posteriori method (Yamamoto 2005); GPI can analyze surface trend of regionalized variables, test the longterm and global trend effect. Based on the sample data, GPI is susceptible to extreme values and is suitable for small surface changes of regionalized variables (Ding et al. 2011; Mutua 2012; Wang et al. 2014); LPI can be used to establish smooth surface, calculate shortterm variability and reflect local variability, but has errors for longdistance global interpolation; weight principle is used by IDW for interpolation analysis, as an accurate interpolation method (Xie et al. 2011), it requires sufficient and welldistributed samples (Rabah et al. 2011). Uneven distribution or outliers may lead to errors; tension spline interpolation (TSPLINE) is an accurate interpolation method (Hofierka et al. 2002) and the interpolation surface goes through every sample point, which is suitable for a large number of data interpolation. The accuracy of which is close to Kriging methods. TSPLINE has an advantage to avoid estimation of covariance function structure.
Interpolation methods have been widely used to analyze spatial variability of precipitation, evapotranspiration, temperature and groundwater level by comparing different interpolation methods. Crossvalidation along with applicable conditions of different models are applied to obtain the bestfit interpolation model (Mardikis et al. 2005; Yang et al. 2011). With the development of GIS technology, interpolation analysis of groundwater level with geostatistical modules has become easer and more operable (Salah 2009; Ta’any et al. 2009; Nikroo et al. 2010), which can characterize the spatial variability of variables in detail (Uyan and Cay 2013; Triki et al. 2013; Dinka et al. 2013; Bao et al. 2014).
Geostatistical method is a good way for analyzing spatial variability of groundwater level by summarizing the previous researches. Based on seven interpolation methods and semivariable function model in GIS geostatistical module, the purposes of the study are (1) compare the prediction accuracies of different methods and select the bestfit interpolation model for piedmont plain area; analyze the spatial variability of the groundwater level in piedmont plain based on hydrogeological conditions and semivariable function, (2) identify the spatial variability characteristics of different hydrogeological units, carry out groundwater level variability partitions considering the land use of piedmont plain and discuss effects of different land use on spatial variability characteristics of groundwater level.
Research method
Survey of study area
The land is mainly used for agriculture. The agricultural area, residential area, industrial area, forest area and the river cover 35.1, 20.5, 10.3, 26.2 and 6.9 % respectively. The water system in this area belongs to Wenyu River basin of the north canal river system. The groundwater regimes type is infiltration—exploitation. Groundwater level shows a seasonal change trend during the year. The drawdown of groundwater level during 2001 and 2013 reaches 22 m and the annual speed of reduction is approximately 2 m.
Thesis data
In order to monitor the dynamic changes of the groundwater level, 30 observation wells (2001–2013) are set up in the area of 550 km^{2} (the plain area and the piedmont area). Quaternary Stata in the piedmont plain is taken as the studying object. Water level measuring tape is used to monitor groundwater level and the observation is made every 5 days (i.e. observations are made on 1st, 6th, 11th, 21th, 26th days of each month). The land use data is collected from the aerial map in 2005, the resolution is 1 m.
Parameters using in the interpolation model
Interpolation model  Parameter P or smooth coefficient^{a}  Maximum number of prediction points within the search radius  Minimum number of prediction points within the search radius  Orientation angle^{b} 

IDW  1.1102  16  9  0 
GPI  2  –  –  – 
LPI  1  14  11  0 
TSPLINE  1  15  10  0 
OK  0.12  13  9  45 
SK  0.138  5  2  0 
UK  1  30  20  45 
Research method
 1.
Construction of basic information database ArcGIS data management is used to obtain the locator and attribute data of groundwater level, land use for each observation well and use them as the basic data for calculation and analysis with geostatistical module.
 2.Evaluation of different interpolation model Inverse distance interpolation (IDW), GPI, LPI, TSPLINE, OK, SK and UK are applied to establish the groundwater level model, mean errors, rootmeansquare errors, coefficient of determinations and absolute errors are calculated for each method to select the optimal model. The computational formulas are as follows (Nikroo et al. 2010; Sun et al. 2009):

1. Mean error

where \(\hat{Z}_{\text{i}}\) is the estimated value, Z _{ i } is the measured value at sampling point i \(({\text{i}} = 1, \ldots ,{\text{n}})\), n is the number of values used for the estimation.$$ME = \frac{1}{n}\left( {Z_{i}  \hat{Z}_{i} } \right)$$(1)


2. Rootmeansquare error

where \(\hat{Z}_{\text{i}}\) is the estimated value, Z _{ i } is the measured value at sampling point i \(({\text{i}} = 1, \ldots ,{\text{n}})\), n is the number of parameters selected by the empirical formula; n is the number of values used for the estimation.$$RMSE = \sqrt {\frac{1}{n}\left( {Z_{i}  \hat{Z}_{i} } \right)^{2} }$$(2)


3. Coefficient of determination

$$R^{2} = \frac{{\left[ {\sum\nolimits_{i = 1}^{n} {\left( {P_{i}  P_{ave} } \right)\left( {Q_{i}  Q_{ave} } \right)} } \right]^{2} }}{{\sum\nolimits_{i = 1}^{n} {\left( {P_{i}  P_{ave} } \right)^{2} \sum\nolimits_{i = 1}^{n} {\left( {Q_{i}  Q_{ave} } \right)}^{2}} }}$$(3)

The coefficient of determination R^{2} is used to measure the correlation between the predicted value and the measured value. P _{ ave } is the averaged estimated value, Q _{ ave } is the averaged measured value; and n is the number of values used for estimation.



 3.Semivariogram, also known as semivariogram, is a unique function of geostatistical analysis. It can be used to describe the spatial variability of groundwater levels. Assume that the mean of the random function is stable and the value is only related to the distance between samples, semivariogram r(h) may be defined as half the incremental variance of random function Z(x).where r(h) is the semivariogram, Z(x) is the random function, h is the distance between samples.$$r(h) = \frac{1}{2}E\left[ {Z(x)  Z(x + h)} \right]^{2}$$(4)

A spherical model of the tested semivariogram models was fitted to the experimental semivariograms. The spherical model is defined by the following equation:where H _{0S } is the nugget value arising from random components such as measurement error and physical factors, H _{ S } is the structural variance arising from spatial autocorrelation, H_{0S} + H _{ S } is the sill, and b is the distance at which the semivariogram equals 95 % of its sill variance.$$\rho (h) = \left\{ {\begin{array}{*{20}l} 0 \hfill &\quad {x = 0} \hfill \\ {{\text{H}}_{0S} + {\text{H}}_{S} \left( {\frac{{3{\text{h}}}}{{2{\text{a}}}}  \frac{{h^{3} }}{{2a^{2} }}} \right)} \hfill &\quad {0 < x \le b} \hfill \\ {{\text{H}}_{0S} + {\text{H}}_{S} } \hfill &\quad {x > b} \hfill \\ \end{array} } \right.$$(5)

 4.
Spatial variability characteristics, is represented by the nugget to sill radio. The nugget value represents the variability while the sill value represents the overall variability inside the variables. The nugget to sill radio shall be between 0 and 1. When the nugget effect is <0.25, the considered strongly spatially dependent; the variable is considered moderately spatially dependent as the nugget effect is between 0.25 and 0.75, while the nugget effect is >0.75, the variable is considered weakly spatially dependent (Ghazi et al. 2014).
 5.
Analysis of spatial distribution pattern: based on the hydrogeological conditions and different land use, the spatial variability pattern of typical partitions in the piedmont plain are analyzed, the effect of human activities on spatial variability of groundwater level is also discussed.
Results and discussion
Model parameter selection and verification
In order to obtain the distribution characteristics of groundwater level, the selection of model parameters shall meet the principle that the mean errors and rootmeansquare errors are close to 0 and 1, respectively. The parameters occur in the Geostatistical Analyst Model of Arcmap 10.3, which are used to verify the model and approach the best interpolation effect for each method. The selected parameters are shown in Table 1.
Interpolation calculation results
The cross validation method is a statistical analysis method used to verify the accuracy of interpolation model, the basic idea is to classify the original dataset into the train set and the validation set. The validation set is used to test the model obtained from the training set, which is the indicators to evaluate the accuracy of the model. The error being the minimum is the evaluation criteria for bestfit interpolation model (Dashtpagerdi et al. 2013).
The statistic of errors in the process of groundwater interpolation in the last 5 years (2009–2013)
Year  Errors  Methods  

2013  Interpolation model  IDW  GPI  LPI  TSPLINE  OK  SK  UK 
Mean error  0.17  −0.73  0.14  0.02  −0.04  0.02  0.03  
Rootmeansquare (m)  0.89  1.46  1.07  1.24  0.26  0.11  0.24  
R^{2}  0.92  0.89  0.91  0.90  0.96  0.99  0.96  
2012  Interpolation model  IDW  GPI  LPI  TSPLINE  OK  SK  UK 
Mean error  0.11  0.79  −0.08  0.04  −0.10  0.03  −0.09  
Rootmeansquare (m)  0.92  1.49  1.10  1.27  0.29  0.04  0.47  
R^{2}  0.88  0.85  0.87  0.86  0.92  0.95  0.92  
2011  Interpolation model  IDW  GPI  LPI  TSPLINE  OK  SK  UK 
Mean error  0.19  −0.71  0.16  −0.04  0.02  −0.01  0.01  
Rootmeansquare (m)  0.17  0.72  0.14  0.02  0.04  0.02  0.03  
R^{2}  0.91  0.88  0.90  0.89  0.95  0.98  0.95  
2010  Interpolation model  IDW  GPI  LPI  TSPLINE  OK  SK  UK 
Mean error  0.19  −0.82  0.18  0.02  0.04  0.04  0.03  
Rootmeansquare (m)  0.85  1.40  1.03  1.19  0.25  0.18  0.42  
R^{2}  0.90  0.87  0.89  0.88  0.94  0.97  0.94  
2009  Interpolation model  IDW  GPI  LPI  TSPLINE  OK  SK  UK 
Mean error  0.14  −0.77  −0.06  0.07  −0.07  0.05  −0.06  
Rootmeansquare (m)  0.90  1.45  1.08  1.24  0.30  0.23  0.47  
R^{2}  0.88  0.86  0.89  0.89  0.92  0.98  0.95 
Interpolation effect comparison among different models
Figure 2 shows the interpolation effect of groundwater level in Beijing piedmont plain from seven interpolation model for 2013. In order to evaluate the simulation results, the “buphthalmos” phenomenon is introduced (Yang et al. 2011). Which is caused by some extreme values in the processing of interpolation, it means the interpolation effect is not good.
As can be seen from Fig. 2, uneven distribution of observation points and the existence of some extreme value points leads to the “buphthalmos” phenomena for IDW and LPI methods (Rajagopalan and Lall 1998) and the interpolation effect is poor; as a function form of radial basis function (RBF) interpolation model, TSPLINE is suitable for establishing a smooth surface with a large amount of data and the interpolation effect is good for undulating surface. However, it is not suitable for the situation where the shortterm variability is high, sample data errors and uncertainties exist. Therefore, the smoothness of the model is poor. GPI considers the global variation trend during the interpolation process and the smoothness is high, which is suitable for spatial variability interpolation whose variability is slow. However, GPI ignores the local variability and its fitting effect of the firstorder model is not good (Johnston et al. 2001b). With the increase of order, it is difficult to explain the precise physical meaning (Apaydin et al. 2004).
As the unbiased optimal interpolation method, Kriging method combines the effects of distance parameters and direction parameters, it represents the spatially continuous and irregular change of variables. Therefore, the interpolation effect of Kriging model is better than others in theory. From the perspective of interpolation results, the SK interpolation is more approximate to the real situation and the groundwater surface looks smooth, which means the interpolation effect is good. Based on the above analysis, SK interpolation method is evaluated as the optimal interpolation model.
Spatial variability analysis of groundwater level
The SK interpolation is used to analyze the spatial variability characteristics of groundwater level between 2001 and 2013 to understand the distribution pattern and variability characteristics of the groundwater level. The spatial variability of the groundwater level is affected by both the natural factors and human factors. The nugget effect reflects the spatial correlation characteristics of the groundwater level (Ahmadi and Sedghamiz 2007; Desbarats et al. 2002). Factors such as precipitation, topographic inequality, aquifer lithology and different hydrogeological units will increase the spatial correlation of groundwater level, while factors such as largescale construction of water conservancy establishments and human exploitation will decrease the spatial correlation.
Semivariance function model parameters of groundwater level (2001–2013)
Year  2001  2002  2003  2004  2005  2006  2007  2008  2009  2010  2011  2012  2013 

Nugget value  79.3  83.5  90.3  81.6  83.8  92.3  116.9  129.6  106.6  97.2  99.2  103.1  107.2 
Partial sill value  135.6  124.9  115.8  105.1  103.9  78.3  63.0  48.9  72.3  87.2  85.6  82.1  78.2 
Sill value  214.9  208.4  206.1  186.7  187.7  170.6  179.9  178.5  178.9  184.4  184.8  185.2  185.4 
Nugget effect  0.37  0.40  0.44  0.44  0.45  0.54  0.65  0.73  0.60  0.53  0.54  0.56  0.58 
Analysis of spatial distribution characteristics
The average annual decline rate of groundwater level for observation wells
Period  Average annual decline rate  Land use  

Agricultural area  Residential area  Industrial area  River area  
2001–2006  0.51  0.35  0.77  0.65  0.25 
2007–2013  2.06  0.65  1.28  2.34  3.95 
Because of overexploitation, the annual average reduction speed during 2007 and 2013 is higher than that during 2001 and 2006. Under the same land use, the change of exploitation way of the groundwater will cause the change of reduction speed of the groundwater level (Choi et al. 2012). Due to the rapid development of the urban area and increase of exploitation amount of the groundwater year by year, the groundwater level during 2006 and 2013 presents an overall downward trend. In the residential area and industrial area, the exploitation quantity of the groundwater is high and the groundwater level reduction speed increases significantly; the agriculture area is mainly distributed in the middle and lower plain area of the alluvial–proluvial fan. The exploitation quantity of the groundwater is of small amount and the reduction speed is small; in the middle and lower river area of the alluvial–proluvial fan, the reduction speed of the groundwater level increases significantly. The reasons are as follows: the severe shortage of the surface water decreases the lateral infiltration recharge quantity; excessive exploitation causes the lateral recharge rate of the piedmont groundwater to be smaller than the groundwater level reduction speed in the river area; the complete lining of the Jingmi diversion canal in the central part of the plain area also aggravates the situation that the groundwater cannot be recharged in time.
Conclusion
 1.
This paper is based on the ArcGIS geostatistical module and semivariable function model to analyze the spatial variability of the groundwater level, compare the simulation accuracies and prediction effects of seven interpolation models and select SK interpolation as the optimal interpolation model for the piedmont plain in Beijing. The results show that the SK interpolation can make unbiased optimal estimation of unknown points while ensuring the local accuracy.
 2.
The spatial variability of the groundwater level is closely related to the hydrogeological units and land use. Owing to the excessive exploitation, the spatial variability of the groundwater level for the top and lower part of the alluvial–proluvial fan is small; the spatial variability of the groundwater level for the middle and upper part of the alluvial–proluvial fan is high; In the residential area, industrial area and river area, the reduction speed of the groundwater level is high; the establishment of new town causes the arable land to decrease and watersaving irrigation decreases the agricultural water quantity, and the reduction rate of the groundwater level is small in the agricultural area.
 3.
The simple Kriging method is applicable for the interpolation of groundwater level in piedmont plain area and the geostatistical interpolation models are of significance in studying the spatial and time variability of the groundwater level. It can also effectively optimize the exploited well distribution and slow down the decline rate of groundwater level.
Declarations
Authors’ contributions
XY have made substantial contributions to conception and design, GXM analyzed and interpreted the data, NY has made contributions to the acquisition of data. SJL and CYL have been involved in drafting the manuscript or revising it critically for important intellectual content; ZQL and YSY have given final approval of the version to be published. All authors read and approved the manuscript.
Acknowledgements
This work was supported by Ministry of Water Resources Public Projects 201101051.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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