Environ Monit Assess (2011) 176:663–676 DOI 10.1007/s10661-010-1611-4
Indicator and probability kriging methods for delineating Cu, Fe, and Mn contamination in groundwater of Najafgarh Block, Delhi, India Partha Pratim Adhikary · Ch. Jyotiprava Dash · Renukabala Bej · H. Chandrasekharan
Received: 19 November 2009 / Accepted: 9 July 2010 / Published online: 5 August 2010 © Springer Science+Business Media B.V. 2010
Abstract Two non-parametric kriging methods such as indicator kriging and probability kriging were compared and used to estimate the probability of concentrations of Cu, Fe, and Mn higher than a threshold value in groundwater. In indicator kriging, experimental semivariogram values were fitted well in spherical model for Fe and Mn. Exponential model was found to be best for all the metals in probability kriging and for Cu in indicator kriging. The probability maps of all the metals exhibited an increasing risk of pollution over the entire study area. Probability kriging estimator incorporates the information about order relations which the indicator kriging does not, has improved the accuracy of estimating the probability of metal concentrations in groundwater being higher than a threshold value. Evaluation of
P. P. Adhikary · Ch. J. Dash · H. Chandrasekharan Indian Agricultural Research Institute, Pusa, New Delhi 110 012, India R. Bej Directorate of Water Management, Bhubaneswar 751 023, Orissa, India Present Address: P. P. Adhikary (B) Research Centre, Central Soil and Water Conservation Research and Training Institute, Datia 475 661, M.P., India e-mail:
[email protected]
these two spatial interpolation methods through mean error (ME), mean square error (MSE), kriged reduced mean error (KRME), and kriged reduced mean square error (KRMSE) showed 3.52% better performance of probability kriging over indicator kriging. The combined result of these two kriging method indicated that on an average 26.34%, 65.36%, and 99.55% area for Cu, Fe, and Mn, respectively, are coming under the risk zone with probability of exceedance from a cutoff value is 0.6 or more. The groundwater quality map pictorially represents groundwater zones as “desirable” or “undesirable” for drinking. Thus the geostatistical approach is very much helpful for the planners and decision makers to devise policy guidelines for efficient management of the groundwater resources so as to enhance groundwater recharge and minimize the pollution level. Keywords Cu · Fe · Mn · Groundwater quality · Semivariogram · Indicator kriging · Probability kriging · Delhi
Introduction India is the seventh largest country in the world having about 16% of world’s population, 2.4% of world’s land, and 4.2% of world’s water resources. Driven by rising population and economic growth in agriculture, industry, and other sectors, India’s
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demand for freshwater is increasing rapidly. The same is the case for Delhi. The population of Delhi is more than 15 million and requires 3,324 million liters of water a day (MLD) to meet its domestic need, while what it gets stands closer to 2,034 MLD. The average water consumption in Delhi is estimated at being 240 l per capita per day (CGWB 2006), the highest in the country. Beside this, huge amount of water is needed for irrigation and other industrial uses. To meet this huge demand of freshwater, groundwater plays a crucial role as a decentralized source of drinking water for millions of rural and urban families. Because of insidious nature of groundwater pollution, it is necessary to know the spatial distribution of polluted groundwater as it takes many years to show its full effect in the quality of water pumped from wells. As the pollutant concentration values are rarely available for every possible location, therefore, geostatistics have focused on prediction of pollutant concentration at any unsampled location. The geostatistical concepts and its applications are reported by different researchers around the world (Isaaks and Srivastava 1989; Goovaerts 1997; Kumar and Ahmed 2003; Stacey et al. 2006; Lu et al. 2007; Lado et al. 2008; Santra et al. 2008; Liu et al. 2009; Orton et al. 2009). Specifically, indicator kriging (Goovaerts 1997), which is a nonparametric approach, is more advanced due to its ability to take the data uncertainty into account and often used to predict the conditional probability for the categorical data of an unsampled location (Goovaerts 1994; Oyedele et al. 1996). In the indicator kriging method, any observation z(x) is assigned to binary indicator code 1 or 0 depending upon whether the observation is greater or smaller than a threshold value zth . The binary indicator codes have no uncertainty as there is no uncertainty about the location of the observation x. So the indicator approach is used without any restriction, and it has become very popular in natural resources studies (Journel 1988; Chiueh et al. 1997; Juang and Lee 1998, 2000; Goovaerts et al. 2005; Lee et al. 2007). However, Journel (1989) pointed out that there will certainly be some measurement errors in observations, as the indicator approach does not consider the uncertainty occurring from measure-
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ment errors. Moreover, indicator kriging is often associated with the problem wherein order relation deviations from the threshold value are not considered (Goovaerts 1997; Juang et al. 1998). Juang and Lee (1998) also pointed out that indicator coding only indicate that z(x) is greater than/lower than/equal to the threshold value zth . It does not take into account information about the order relation of z(x). The indicator function under the threshold value zth only describes the spatial variability of attribute values close to zth . Therefore, using indicator kriging to estimate the probability of z(x) being greater than or less than zth may result in loss of the spatial information of the attribute values which are not close to zth (Juang and Lee 2000). In order to improve indicator kriging estimation, indicator functions under various thresholds differing from the desired threshold zth have been used via the indicator cokriging estimator (Goovaerts 1994). In indicator cokriging, these indicator functions under various thresholds are assigned as auxiliary variables to improve estimation of the probability of attribute values being greater/lower than zth . The indicator functions, defined under various thresholds other than zth and used as auxiliary variables, represent the spatial variability of attribute values, which are not close to zk . Thus, the indicator cokriging estimator can include information from all available attribute values and may be better than the indicator kriging estimator. However, in practice, indicator cokriging offers little improvement over indicator kriging (Goovaerts 1994) as the indicator cokriging is more complex and cumbersome when the number of indicator functions corresponding to auxiliary variables is greater than one. Thus, another nonparametric approach, called probability kriging which is also based on cokriging estimator has also been proposed. This method represents the information from all the available attribute values by using the order relation of observed values (Carr and Mao 1993; Carr 1994). The uniform value, also called the standardized rank, which denotes the order relation of observed values, is assigned as the only auxiliary variable in probability kriging to improve estimation of the probability of the attribute value being greater/lower than the desired
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threshold. Thus, probability kriging may be better than indicator kriging and also less complex and cumbersome. The probability kriging has been shown to be better than indicator kriging for delineating contaminated soils (Goovaerts 1997; Juang and Lee 2000). However, the performance of probability kriging has not been compared with that of indicator kriging for contaminated groundwater. Thus, the focus of this study was to compare two nonparametric kriging methods such as indicator and probability kriging and to judge the best kriging method to delineate the area where the Cu, Fe, and Mn concentration in the groundwater exceeds the threshold values of drinking water standard. Materials and methods Study area Main focus of this study was to quantitatively assess the Cu, Fe, and Mn concentration in ground-
Fig. 1 Map of study area showing locations of observation wells in Najafgarh Block, Delhi, India
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water of Najafgarh block, Delhi, India, which lies between north latitude of 28◦ 30 14.13 and 28◦ 39 44.46 and east longitude of 76◦ 50 24.03 and 77◦ 02 14.81 and covers an area of about 189 km2 . The Najafgarh drain makes the southern and eastern boundary and Kultana Chhudani Bupania (KCB) drain makes the northern boundary of the study area (Fig. 1). The climate is subtropical and semi arid with average annual rainfall of about 611.8 mm. About 85% of annual rainfall is received from July to September. Soils of the study area belong to sandy loam to loamy sand textural classes while in some minor pockets loam texture is evidenced. Most of the soils come under Palam series which comprises of very deep, yellowish brown alluvial soils. The land use pattern is dominated by agriculture, comprises about 75% of total area and the rest is occupied by habitats, roads, ponds, forests etc. Tube well is the main source of irrigation which is deep, bore because of very low groundwater table (beyond 20–30 m; CGWB 2006).
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Groundwater sampling and measurement
Kriging techniques
Ninety-three groundwater samples were collected from the existing tube wells, hand pumps, and open wells in the study area. Out of these 93 samples, 65 were from tube wells, 23 from hand pumps, and five from open wells. These sampling locations are shown in Fig. 1. Sampling site geo-positions (latitude and longitude) were determined using a global positioning system. Collected samples were analyzed in the laboratory to measure the concentration of the Cu, Fe, and Mn using atomic absorption spectrophotometer. The software GS+ 8 (Gamma Design Software, LLC, Michigan, USA 1988) was used for geostatistical analysis of data and Arc GIS 9.3 (ESRI, New York, USA 2004; Johnston et al. 1996) was used for generation of maps.
In kriging method, any parameter can be predicted at the unsampled location (x0 ) within the system domain (D) using information available elsewhere in D(x1 , x2 , x3 . . . .xn ). This can be carried out by expressing Z (x0 ) {where Z (x): xε D) as a linear combination of the data {Z (x1 ), Z (x2 ),. . . Z (xn )}, such that: λi Z (x0 )
(1)
Semivariance
a Copper concentration
Separation distance (km)
b Iron concentration Semivariance
The concentrations of Cu, Fe, and Mn in groundwater were analyzed to get the descriptive statistics of each parameter. The normality of each data set was checked by Kolmogorov–Smirnov test and different transformations such as log normal, square root were carried out to ensure normal distribution. To perform indicator kriging, the groundwater Cu, Fe, and Mn concentrations were transformed into indicator codes using Eq. 3, and to perform probability kriging into uniform values using Eq. 7. Then geostatistical software GS+ was used to generate the semivariogram parameters for each theoretical model such as spherical, exponential etc. The best-fitted theoretical model was selected based on the highest R2 . The corresponding sill, nugget, and range values of the best-fitted theoretical models were observed (Figs. 2 and 3). Subsequently, probability of exceedance maps for Cu, Fe, and Mn concentration in groundwater were generated using indicator and probability kriging based on the threshold values of the pollutants in drinking water. In this study the threshold values of the Cu, Fe, and Mn concentrations for declaring groundwater as unsafe for drinking were set to be 1.5, 1.0, and 0.5 mg/l, respectively, as per the standard of BIS (1991).
n i=1
Separation distance (km)
Semivariance
Descriptive statistics and threshold limits
∧
Z (x0 ) =
c Manganese concentration
Separation distance (km)
Fig. 2 a–c Best-fitted semivariograms of indicator codes for Cu, Fe, and Mn concentrations in groundwater
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where λi is the kriging weight of the parameter value at Z (x0 ) for n nearby sample points to be used in estimation.
Semivariogram f itting Semivariogram is used to quantify the spatial structure of the variables. It is the graphical representation of the mean square variability between
Semivariance
a Copper concentration
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two neighboring points of distance h and represented in Eq. 2. N(h) 2 1 γ (h) = z (xi + h) − z(xi ) 2 N(h) i=1
(2)
where γ (h) is the semivariogram and is expressed as a function of the magnitude of the lag distance, N(h) is the number of observation pairs for xi and (xi + h) and z(xi ) is the variable at location xi . The experimental semivariogram γ (h) can be fitted to different theoretical models such as Spherical, Exponential, Linear, or Gaussian to determine three semivariogram parameters, such as nugget (C0 ), sill (C0 + C), and range (A0 ; Isaaks and Srivastava 1989). Semivariograms can be computed in different directions to detect any anisotropy of the spatial variability. Here only isotopic variations were considered. Indicator kriging
Separation distance (km)
Semivariance
b Iron concentration
In the indicator kriging procedure, first the indicator codes are generated by the indicator function, which is under a desired threshold value zth . It is written as follows: 1, if z(x) ≥ zth I (xi ; zth ) = (3) 0, otherwise Then, the semivariogram γi (h) is used to quantify the spatial correlation of the indicator codes, I(xi ; zth ),and it is written as follows:
Separation distance (km)
Semivariance
c Manganese concentration
γi (h) =
N(h) 2 1 I(xi ; zth ) − I(xi + h ; zth ) 2N(h) i=1
(4) The indicator kriging estimator, I ∧ (x0 ; zth ) at the location x0 can be calculated by I ∧ (x0 ; zth ) =
n
λi I(xi ; zth )
i=1
and the indicator kriging system given Separation distance (km)
Fig. 3 a–c Best-fitted semivariograms of uniform values for Cu, Fe, and Mn concentrations in groundwater
n j=1
λ jγi (xj − xi ) = γi (xo − xi ) − μ
(5)
λi = 1 is
(6)
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biased conditions λij = 1 and λup = 0, unknown weights can be solved from the resulted cokriging system as follows:
where λ j is the weighted coefficient, γi is the semivariance of the indicator codes at the respective lag distance, and μ is the Lagrange multiplier.
n
Probability kriging
λijγi (xj − xi ) +
U(x) ≈
r n
= γi (x0 − xi ) − μi n
The geostatistical software GS+ was used to generate the semivariogram parameters for each theoretical model such as Spherical, Exponential, and Gaussian using indicator codes and uniform values. The best-fitted theoretical model was selected based on the highest R2 . The corresponding sill, nugget, and range values of the best-fitted theoretical model were observed. Subsequently, the probability of exceedance maps for Cu, Fe, and Mn were generated using indicator kriging and probability kriging based on their threshold values in drinking water. The indicator kriging and probability kriging were performed using the geostatistics extension module of ArcGIS 9.3. A comparison of the performance of two nonparametric kriging methods was performed based on the mean error (ME), mean squared error (MSE), kriged reduced mean error (KRME), and kriged reduced mean squared error (KRMSE)
N(h) 1 {{I(xi ; zth ) − (xi + h; zth )} 2N(h) i=1
I (x0 ; zth ) =
n
λi I (xi ; zth ) +
i=1
n
λui U(xi )
(9)
(10)
i=1
where λi and λui are the weights associated with I(xi ; zth ) and U(xi ). To minimize the variance of the estimation error and based on the un-
Table 1 Descriptive statistics of groundwater Cu, Fe, and Mn across the study area
(12)
Selection of best-f it theoretical model, semivariogram parameters, and cross-validation
The probability kriging estimator is defined by ∧
λup γu (x p − xi )
p=1
where γi , γu , and γiu are the semivariance and cross-semivariance of I(xi ; zth ) and U(xi ), and μi and μu are the Lagrange multipliers, respectively, for I(xi ; zk ) and U(xi ).
(8)
× {U(xi ) − U(xi + h)}}2
n
= γiu (x0 − xi ) − μu
Moreover, a cross-semivariogram used to show the cross-spatial correlation between I(xi ; zth ) and U(x) is γiu (h) =
λijγiu (xj − xi ) +
j=1
(7)
N(h) 2 1 U(xi ) − U(xi + h) 2N(h) i=1
(11)
and
where r denotes the rank of the rth order statistic z(r) located at x and n is the total number of observations (Goovaerts 1997). The semivariogram of U(x) is γu (h) =
λup γiu (x p − xi )
p=1
j=1
For probability kriging, the indicator code I(xi ; zth ) is assigned as the main variable and the other variable, the uniform value U(x), is assigned as the auxiliary variable in the cokriging estimator. The uniform value, also called the standardized rank, was reported in detail by Journel and Deutsch (1997) and is defined as
n
Groundwater No. of Minimum Maximum Mean Skewness Kurtosis Standard parameters observation (mg/l) (mg/l) (mg/l) deviation (mg/l) Cu Fe Mn
93 93 93
0.4 0.3 0.3
3.2 3.1 3.0
1.61 1.39 1.34
0.14 0.62 0.54
2.19 2.66 2.17
0.663 0.673 0.718
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5.0 5.0 5.0 0.107 0.107 0.107 0.029 0.029 0.029 Exponential Exponential Exponential 0.667 0.558 0.871 41.00 41.00 1.83 0.413 0.353 0.057 0.206 0.176 0.009
Range (A0 ) Sill (c0 +c) Nugget (c0 )
Uniform values
Best-fitted model R2 Range (A0 ) Sill (c0 +c) Nugget (c0 ) Best-fitted model
Indicator codes
Exponential Spherical Spherical 0.680 0.920 0.979 2.65 1.10 7.30 0.558 0.474 0.564 0.229 0.001 0.193
The descriptive statistics of Cu, Fe, and Mn concentrations in the groundwater across the study area are listed in Table 1. The Cu concentration in groundwater ranged from a minimum of 0.4 to a maximum of 3.2 mg/l with 55 numbers of wells (59%) out of 93, having higher concentration than the permissible level (1.5 mg/l). The Fe concentration ranged from 0.3 to 3.1 mg/l and the mean value was 1.39 mg/l. The Fe concentration exceeded the drinking water criterion value, i.e., 1.0 mg/l for 60 wells (65%). Similarly, the Mn concentrations ranged from a minimum of 0.3 mg/l to a maximum of 3.0 mg/l. There were 88 wells (95%) that exceeded the permissible limit of Mn
Exponential Exponential Spherical
Distribution pattern of Cu, Fe, and Mn concentrations in groundwater
Cu Fe Mn
Results and discussion
R2
where zo,i is the observed value at location i, z p,i is the predicted value at location i and N is the number of pairs of observed and predicted values. s is the standard deviation of the observed values. The mean error and kriged reduced mean error values near to zero is an indicator of better model prediction. As a practical rule, the MSE should be less than the variance of the sample √ values and KRMSE should be in the range 1 ± (2 2)/N.
Range (A0 )
(16)
Sill (c0 +c)
N 1 (zo,i − z p,i )2 ∼ KRMSE = =1 N i=1 s2
(15)
Nugget (c0 )
N 1 (zo,i − z p,i ) ∼ =0 N i=1 s
Original values
KRME =
(14)
Best-fitted model
N 1 (zo,i − z p,i )2 ∼ MSE = = minimum N i=1
(13)
Groundwater parameters
N 1 (zo,i − z p,i ) ∼ = 0 N i=1
Table 2 Semivariogram parameters of the best-fitted theoretical models for original values, indicator codes, and uniform values
ME =
R2
generated by the cross-validation procedure (Sarangi et al. 2005). The ME, MSE, KRME, and KRMSE of the indicator kriging estimates and probability kriging estimates were calculated accordingly:
0.946 0.946 0.946
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b) Fe concentration
a) Cu concentration
c) Mn concentration Fig. 4 a–c Spatial variability maps of Cu, Fe, and Mn concentration in groundwater in the study area
concentration (0.5 mg/l) as deemed fit for drinking in India as per BIS (1991). The mean and variance of uniform values for all the three metals under consideration were found to be equal as they only
dependent upon the sample size (N = 93). From Kolmogorov–Smirnov test, it was observed that all the data sets were normally distributed at 5% level of significance.
Table 3 Delineated area under different concentration limits of groundwater Cu, Fe, and Mn in the study area
Fe
Cu
Mn
Concentration limits (mg/l)
Area (km2 )
Concentration limits (mg/l)
Area (km2 )
Concentration limits (mg/l)
Area (km2 )
0.4–1.0 1.0–1.5 1.5–3.2
2.5 54.4 132.2
0.3–1.0 1.0–1.5 1.5–3.1
4.0 143.6 41.4
0.3–0.5 0.5–1.5 1.5–3.0
0 118.7 70.3
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Spatial structure of groundwater quality parameters The best-fitted theoretical models and their corresponding semivariogram parameters of groundwater Cu, Fe, and Mn concentrations for indicator codes are shown in Fig. 2. The nugget, sill, and range values of the best-fitted theoretical models for the metals under study are shown in Table 2. It
was observed that the range parameter in the bestfitted theoretical model of Mn was lowest, where as Cu and Fe exhibited same higher range value. The exponential semivariogram model was observed to be the best-fitted model for Cu, whereas the spherical model fitted well for Fe and Mn. For uniform values, the best-fitted theoretical models and their corresponding semivariogram parameters are shown in Fig. 3 and presented in Table 2.
a) Cu concentration
b) Fe concentration
c) Mn concentration Fig. 5 a–c Probability maps of Cu, Fe, and Mn concentration in groundwater in the study area based on indicator kriging
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Table 4 Delineated area under different probability ranges of threshold concentration limits of groundwater Cu, Fe, and Mn in the study area obtained using indicator kriging
Cu
Fe
Mn
Probability range
Area (km2 )
Probability range
Area (km2 )
Probability range
Area (km2 )
0.0–0.2 0.2–0.4 0.4–0.6 0.6–0.8 0.8–1.0
0.3 34.5 104.0 39.6 10.5
0.0–0.2 0.2–0.4 0.4–0.6 0.6–0.8 0.8–1.0
0.4 11.2 66.4 75.7 35.3
0.6–0.7 0.7–0.8 0.8–0.9 0.9–0.95 0.95–1.0
2.3 8.4 88.7 32.7 57.0
In this case, all the parameters fitted to exponential model, and were having same semivariogram parameters. Furthermore, the semivariogram pa-
rameters of the indicator codes and uniform values were used in indicator and probability kriging, respectively, for generation of maps. To ascertain
a) Cu concentration
b) Fe concentration
c) Mn concentration Fig. 6 a–c Probability maps of Cu, Fe, and Mn concentration in groundwater in the study area based on probability kriging
Environ Monit Assess (2011) 176:663–676 Table 5 Delineated area under different probability ranges of threshold concentration limits of groundwater Cu, Fe, and Mn in the study area obtained using probability kriging
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Cu
Fe
Mn
Probability range
Area (km2 )
Probability range
Area (km2 )
Probability range
Area (km2 )
0.0–0.2 0.2–0.4 0.4–0.6 0.6–0.8 0.8–1.0
0.3 35.2 104.0 41.9 7.6
0.3–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–1.0
10.3 42.4 91.4 43.3 1.5
0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1.0
1.5 4.2 21.5 56.2 105.5
the reproducibility of observed value by the theoretical model and developed map, the error statistics were estimated. Spatial variation of groundwater Cu, Fe, and Mn The thematic map of Cu (Fig. 4a) showed that a major portion of the study area has Cu concentration higher than the safe limit of drinking water standard. The safe limit of Cu concentrations is considered to be less than 0.05 mg/l, and the unsafe limit exceeds 1.5 mg/l. As the minimum concentration of Cu in groundwater found to be 0.4 mg/l and maximum is that of 3.2 mg/l, the thematic map of Cu concentration was classified under three ranges such as 0.4–1.0, 1.0–1.5, and 1.5–3.2 mg/l. It was observed from the variability map that about 132.2 km2 of area was unsafe (Table 3). Similarly, the thematic maps of Fe and Mn were generated using the corresponding best-fitted model and semivariogram parameters (Fig. 4b–c). The thematic map of Fe was classified into three as per BIS specifications. It was observed from the map of Fe (Fig. 4b) that the Fe concentration was higher in the central part and south-eastern boundary of the study area and covered an area of only 41.4 km2 . The reason for this may be attributable to the presence of some industry adjacent to Najafgarh city and use of highly polluted water of Najafgarh drain for irriga-
tion. The thematic map of Mn was classified under three ranges such as 0.3–0.5, 0.5–1.5, and 1.5– 3.0 mg/l and corresponding areas are presented in Table 3. No area was found to be under safe zone and about 70.3 km2 out of 189 km2 of study area was having highest Mn values restricting its use for drinking. Comparison of kriging methods The probability maps generated using the indicator kriging method is shown in Fig. 5. The probability map of Cu (Fig. 5a) showed that high copper concentration was evidenced in the southern part of the area. It was also found from the Table 4 that about 10.5 km2 of area is very much vulnerable for drinking purpose, for which the probability of exceedance from the threshold value of Cu was the highest (0.8 to 1.0). This may be attributed to the use of highly polluted Najafgarh drain water for irrigation. Similarly, from the probability map of Fe (Fig. 5b) and Table 4, it was found that about 35.3 km2 of area covered highest probability of exceedance from the threshold value, but it has got no specific trend. Although only 18.7% of the area has shown highest probability of exceedance for iron but more than 40% of the study area covers probability of exceedance range of 0.6–0.8, which is very alarming. For Mn, about 57.0 km2 of area occupied the highest probability of exceedance
Table 6 Cross-validation results for kriging methods ME Cu Fe Mn
MSE
KRME
KRMSE
IK
PK
IK
PK
IK
PK
IK
PK
0.010 0.003 0.009
0.009 0.006 0.009
0.495 0.471 0.312
0.490 0.433 0.310
0.019 0.011 0.031
0.012 0.009 0.025
0.986 0.963 1.022
0.995 0.990 1.012
ME mean error, MSE mean square error, KRME kriged reduced mean error, KRMSE kriged reduced mean square error, IK indicator kriging, PK probability kriging
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from the threshold value of respective pollutant concentration in groundwater. Especially for Mn, the entire study area is showing a very high probability of exceedance from cutoff value. So the corrective measures should be taken to check the Mn pollution in groundwater in an urgent basis. Similarly, the probability maps generated using the probability kriging method is shown in Fig. 6. The trend in the probability map of Cu concentration derived by probability kriging (Fig. 6a) was similar to that of probability map obtained by
indicator kriging. The high Cu concentration was found in the southern part of the study area. It was also found from the Table 5 that about 7.6 km2 of area is vulnerable for drinking water, for which the probability of exceedance from the threshold value of Cu was the highest that is from 0.8 to 1.0. Likewise, from the probability map of Fe (Fig. 6b) and the Table 5, it was found that only a minor pocket, contributing an area of only 1.5 km2 was having highest probability of exceedance over the threshold value, which seems to differ from the
b) Fe concentration
a) Cu concentration
c) Mn concentration Fig. 7 a–c Standard error maps of Cu, Fe, and Mn concentration in groundwater in the study area based on probability kriging
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result obtained by using indicator kriging. For Mn, maximum area, which was about 161.7 km2 having highest probability of exceedance from the groundwater pollution cutoff value and the whole study area is showing very high probability of exceedance from the threshold value. This is quite similar to the results obtained by indicator kriging method. Thus the probability of metal concentrations being higher than the threshold value at an unsampled location can be used to indicate the possibility of contamination. A decision maker can use the probability to determine whether or not unsampled locations or areas require cleanup and, at the same time, obtain information about uncertainty and use it to avoid making incorrect decisions. For all the metals, the error statistics such as ME, MSE, KRME, and KRMSE were estimated and presented in Table 6. It was observed that for all groundwater quality parameters, the error terms ME, KRME were close to zero and KRMSE were close to one. The estimated mean error values of the two kriging methods were found to be very similar. However, the MSE values obtained using probability kriging is less than that of indicator kriging. This indicates that the estimated probabilities obtained using probability kriging had greater variability. Similarly, the KRME and KRMSE values obtained by probability kriging were found to be lower than the values obtained from indicator kriging. This indicates that the accuracy of the probability kriging estimations is higher than that of the indicator kriging estimations which is on an average 3.52% better. As the probability kriging is better than indicator kriging to predict metal pollution risk in the groundwater, the standard error maps of the probability kriging for Cu, Fe, and Mn were generated and presented in Fig. 7. It is clear that for all the metals, the error term is lowest at the central part of the study area and at the boundary it is highest. This may be because of boundary effect. The error is highest for Cu (0.496–0.541) and lowest for Mn (0.307–0.357) because the pollution load in groundwater due to Mn is more than that of Cu. The results suggest that incorporation of the information about order relations can improve the accuracy of estimating the probability of metal
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concentrations in the groundwater being higher than a threshold value. Therefore, probability kriging is more suitable than indicator kriging for estimation of risk from metal pollution in groundwater.
Conclusions The cross-validation results showed that the probability kriging estimations were more accurate than the indicator kriging estimations. This suggests that the probability kriging method, which incorporates the information about order relations, can improve the accuracy of estimating the probability of metal concentrations being higher than a threshold value. The thematic maps generated using the ordinary kriging method showed the variation of different metals concentrations over the study area. The delineated zones with potential pollutant concentration were observed mostly in the southern and eastern part of the study area. In particular Mn concentration in the groundwater was found to be exceeded the safe level over the entire study area. So the direct use of groundwater for drinking in this region should generally be avoided. Therefore it is time to devise policy guidelines for efficient management of the groundwater resources so as to enhancing groundwater recharge and minimizing the pollution level.
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