Q- Q plot (Zn)
4.5 Geostatistical Analysis
4.5.2 Deterministic Methods
4.5.2.3 Radial Basis Function
Radial basis functions (RBF) is the name given to a large family of exact interpolators which use a basic equation dependent on the distance between the interpolated point and the sampling points (Brovelli et al., 2011). In this study, RBF analysis, cross validation of RBF with five kernal function of thin-plate spline (TPS), spline with tension (SPT), completely regularized spline (CRS), multi-quadratic function (MQ) and inverse multi- quadratic function (IMQ) was performed for each metal elements and hence discussed in the following sections. The cross validation result of metal elements were reported in the Annex-E, except for the metal elements such as Cd, Ni, Pb and Zn which is provided in this chapter. When kernal function showed higher value of the kernel parameter, it will provide a smoother surface except IMQ. When kernal function showed lower value of the kernel parameter, it will provide a smoother surface in case of IMQ (ESRI, 2001; Zhao et al., 2010; Yao et al., 2013). A researcher Brovelli et al. (2011) stated that when RMSPE showed the lowest value, the interpolation technique was considerd as best fitted model.
112
CRS ST MQ IMQ TPS
Explanation for CRS
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large green color
portion with
noticeable yellow region.
Explanation for ST
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large green color
portion with
noticeable yellow region.
Explanation for MQ
Comparatively more contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large green color
portion with
noticeable yellow region.
Explanation for IMQ
Comparatively lowest adulteration was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large green color
portion with
noticeable yellow region.
Explanation for TPS
Comparatively higher contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large green color
portion with
noticeable yellow region.
Color variation and
visual contamination
level from spatial distribution
Figure 4.22: Spatial distribution of Cd in soil using RBF’s.
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Legend Cd
Prediction Map [Cd_RBF].[Cd]
Filled Contours 1.2 – 1.87 1.87 – 2.48 2.48 – 3.02 3.02 – 3.52 3.52 – 3.97 3.97 – 4.47 4.47 – 5.01 5.01 – 5.62 5.62 – 6.29 6.29 – 7.03
113 Cross Validation of Cadmium
In RBF analysis, cross validation of RBF with five different kernal functions of CRS, ST, MQ, IMQ and TPS was performed for Cd to find the best fitted model for showing spatial distribution and provided in Table 4.17. Based on the interpolation analysis, it was observed RBF for IMQ showed comparatively the lower value of RMSPE (1.2054) than that of other kernal functions. Based on this finding, IMQ was chosen as the best fitted model for Cd. In case of Cd, the kernal parameter of IMQ showed 0.000121 indicated a very small value tends to zero thus it provided a smoother surface. In addition, the kernal function of TPS showed comparatively the highest value of kernel parameter and provided a smoother surface. Figure 4.18 showed the spatial distribution of Cd for distinct RBF’s.
The produced prediction surface area for TPS, there exhibited maximum greenish color region with Cd concentration approximately ranges from 1.87 to 3.52 mg/kg. In addition, RBF with TPS exhibited maximum red and orange color with Cd concentration combindly ranges from 4.47 to 7.03 mg/kg, indicated the soil of the study area was highly contaminted by Cd (Figure 4.22).
Table 4.17: Results of cross validation of RBF with different kernal functions for Cd
Model CRS ST MQ IMQ TPS
aMPE 0.076 0.075 0.068 0.115 0.122
bRMSPE 1.2239 1.2241 1.3372 1.2054 1.6159
Kernel Parameter 55181.22 43835.15 0 0.000121 1.00E+20
aMPE=Mean Prediction Error, bRMSPE= Root Mean Square Prediction Error
Cross Validation of Nickel, Lead and Zinc
In case of Ni, Pb and Zn, the cross validation of RBF with five different kernal functions of CRS, ST, MQ, IMQ and TPS was performed to find the best fitted model and the results are provided in Table 4.18. The kernal function IMQ was chosen as best fitted model for RBF interpolation in case of Ni. It was found from the cross validation result, RMSPE (0.115) was comparatively lower than that of other kernal functions. The kernal parameter of IMQ showed 0.00013 indicated a very small value tends to zero thus it provided a smoother surface. The kernal function of TPS showed comparatively the highest value of kernel parameter and also provided a smoother surface.
114
CRS ST MQ IMQ TPS
Explanation for CRS
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for ST
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for MQ
Comparatively more contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for IMQ
Comparatively lowest adulteration was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for TPS
Comparatively higher contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Color variation and
visual contamination
level from spatial distribution
Figure 4.23: Spatial distribution of Ni in soil using RBF’s.
Legend Ni
Prediction Map [Ni_RBF].[Ni]
Filled Contours 1.08 – 2.05 2.05 – 2.79 2.79 – 3.35 3.35 – 3.77 3.77 – 4.08 4.08 – 4.51 4.51 – 5.06 5.06 – 5.79 5.79 – 6.77 6.77 – 8.06
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115
Table 4.18: Cross validation of RBF for Ni,Pb and Zn Metals Model aMPE bRMSPE Kernel Parameter
Ni
CRS 0.147 1.204 35188.1
ST 0.144 1.206 35234.7
MQ 0.097 1.294 0
IMQ 0.211 1.197 0.00013
TPS 0.161 1.586 1.00E+20
Pb
CRS 0.736 11.999 91048.6
ST 0.738 11.999 71818.2
MQ 0.477 12.830 0
IMQ 1.187 12.139 0.00011
TPS 0.891 15.644 1.00E+20
Zn
CRS 0.613 7.866 20297.9
ST 0.536 7.915 18199.2
MQ 0.502 8.336 0
IMQ 0.713 7.745 0.00016
TPS 0.849 9.500 1.00E+20
aMPE=Mean Prediction Error, bRMSPE= Root Mean Square Prediction Error IMQ was chosen as best fitted model for RBF interpolation in case of Ni. It was found from the cross validation result, RMSPE (1.197) was comparatively lower than that of other kernal functions. The kernal parameter of IMQ showed 0.00013 indicated a very small value tends to zero thus it provided a smoother surface. The kernal function of TPS showed comparatively the highest value of kernel parameter and provided a smoother surface. Figure 4.23 showed the spatial distribution of Ni for different RBF’s. The produced prediction surface area for TPS, maximum greenish color region with Ni concentration approximately ranges from 1.08 to 4.08 mg/kg. In addition, RBF with TPS exhibited yellow color with concentration of 4.08 to 4.51 mg/kg. Maximum ocher, orange, orange red and red color with Ni concentration combindly ranges from 4.08 to 8.06 mg/kg, which indicated the soil of the study area was moderately contaminted in case of Ni (Figure 4.23). From Table 4.18 for Pb, it was observed RBF for CRS showed comparatively the lower value of RMSPE (11.999) than that of other kernal functions.
Thus, CRS was chosen as the best fitted model for Pb. In case of Pb, the kernal parameter of CRS showed 91048.62 indicated a large value tindicated a smoother produced surface. The kernal function of TPS showed comparatively the highest value of kernel parameter and provided more smoother surface. Figure 4.24 showed the spatial distribution of Pb using distinct RBF’s. The produced prediction surface area for TPS, there exhibited maximum greenish color region with Pb concentration approximately ranges from 10.88 to 33.06 mg/kg.
116
CRS ST MQ IMQ TPS
Explanation for CRS
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for ST
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for MQ
Comparatively more contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for IMQ
Comparatively lowest adulteration was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for TPS
Comparatively higher contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Color variation and
visual contamination
level from spatial distribution
Figure 4.24: Spatial distribution of Pb in soil using RBF’s.
Legend Pb
Prediction Map [Pb_RBF].[Pb]
Filled Contours 10.88 – 16.17 16.17 – 20.33 20.33 – 23.61 23.61 – 27.77 27.77 – 33.06 33.06 – 39.78 39.78 – 48.33 48.33 – 59.19 59.19 – 72.99 72.99 – 90.55
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117
In addition, RBF with TPS exhibited maximum ocher, orange,orange-red and red color with Pb concentration combindly ranges from 39.78 to 90.55 mg/kg, which indicated the soil of the study area was moderately contaminted in case of Pb (Figure 4.24).
In addition, from the cross validation result of RBF, RMSPE (7.745) was comparatively lower for IMQ than that of other kernal functions. The kernal parameter of IMQ showed 0.00016 indicated a very small value tends to zero thus it provided a smoother surface. The TPS showed comparatively the highest value of kernel parameter and provided a smoother surface. The produced prediction surface area for TPS, maximum greenish color region with Zn concentration approximately ranges from 11.82 to 23.8 mg/kg (Figure 4.18). In addition, RBF with TPS also exhibited maximum red and orange color with Zn concentration combindly ranges from 31.39 to 50.76 mg/kg, which indicated the soil of the study area was moderately contaminted in case of Zn (Figure 4.25). Figure 4.23, Figure 4.24 and Figure 4.25 displayed IMQ provided better field condition for metal elements of Ni, Pb and Zn, respectively in soil of waste disposal site. It was comprehended from the analysis that the contamination hotspots were near the center of the selected disposal site for metal element of Ni, Pb and Zn.
In addition, the cross validation of RBF with five distinct functions was performed and the results are provided in Table E.1 to Table E.17 as well as the spatial distribution for metal elements of Al, As, Ba, Ca, Co, Cr, Cu, Fe, Hg, K, Mn, Na, Sb, Sc, Sr, Ti, and V in soil of the study area is depicted in Figure E. 1 to Figure E.17 in Annex-E. Results reveals that in produced prediction surface for all these metal elements,, greenish color region represented the level of less contamination and redish color area represented the level of highly contaminated soil.
118
CRS ST MQ IMQ TPS
Explanation for CRS
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for ST
Comparatively less contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for MQ
Comparatively more contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for IMQ
Comparatively lowest adulteration was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Explanation for TPS
Comparatively higher contamination was visible between functions.
Most of the region exhibited orange red and ocher color.
Small red color region was also visible.
Large yellow color
portion with
noticeable green region.
Color variation and
visual contamination
level from spatial distribution
Figure 4.25: Spatial distribution of Zn in soil using RBF’s.
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Legend Zn
Prediction Map [Zn_RBF].[Zn]
Filled Contours 11.82 – 14.47 14.47 – 17.33 17.33 – 20.44 20.44 – 23.8 23.8 – 27.45 27.45 – 31.39 31.39 – 35.67 35.67 – 40.31 40.31 – 45.32 45.32 – 50.76
119 4.5.3 Geostatistical Methods
Geostatistical techniques assume that at least some of the variation observed in natural phenomena can be modeled by random processes with spatial autocorrelation and require that the spatial autocorrelation be explicitly modeled. Geostatistical techniques can be used to describe and model spatial patterns (variography), predicts values at unmeasured locations (kriging), and assess the uncertainity associated with a predicted value at the unmeasured locations (kriging). The geostatistical wizard offers several types of kriging, which are suitable for different types of data and have different underlying assumption of Ordinary, Simple, Universal, Indicator, Probability, Disjunctive, Areal interpolation etc..
These mean standardized prediction error (MSPE), root mean square standard prediction error (RMSSPE) and average standard prediction error (ASPE) was used to select the best fitted models.