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Journal of Environmental Quality 30:894-903 (2001)
© 2001 American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America

TECHNICAL REPORT
Heavy Metals in the Environment

Using Rank-Order Geostatistics for Spatial Interpolation of Highly Skewed Data in a Heavy-Metal Contaminated Site

Kai-Wei Juanga, Dar-Yuan Leea and Timothy R. Ellsworthb

a Graduate Institute of Agricultural Chemistry, National Taiwan Univ., Taipei, 106 Taiwan
b Dep. of Natural Resources and Environmental Sciences, Univ. of Illinois, Urbana–Champaign, IL 61801

Corresponding author (dylee{at}ccms.ntu.edu.tw)

Received for publication March 6, 2000.

   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The spatial distribution of a pollutant in contaminated soils is usually highly skewed. As a result, the sample variogram often differs considerably from its regional counterpart and the geostatistical interpolation is hindered. In this study, rank-order geostatistics with standardized rank transformation was used for the spatial interpolation of pollutants with a highly skewed distribution in contaminated soils when commonly used nonlinear methods, such as logarithmic and normal-scored transformations, are not suitable. A real data set of soil Cd concentrations with great variation and high skewness in a contaminated site of Taiwan was used for illustration. The spatial dependence of ranks transformed from Cd concentrations was identified and kriging estimation was readily performed in the standardized-rank space. The estimated standardized rank was back-transformed into the concentration space using the middle point model within a standardized-rank interval of the empirical distribution function (EDF). The spatial distribution of Cd concentrations was then obtained. The probability of Cd concentration being higher than a given cutoff value also can be estimated by using the estimated distribution of standardized ranks. The contour maps of Cd concentrations and the probabilities of Cd concentrations being higher than the cutoff value can be simultaneously used for delineation of hazardous areas of contaminated soils.

Abbreviations: CDF, cumulative distribution function • EDA, exploratory data analysis • EDF, empirical distribution function • ME, mean error • MSRE, mean square relative error


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
GEOSTATISTICAL interpolation (kriging) provides the best linear unbiased prediction for spatially dependent properties. Recently, kriging was used for the spatial interpolation of pollutants in contaminated soils (Arrouays et al., 1996). However, the great variability of pollutant distributions in soils in conjunction with sparse sampling will mask the spatial dependence. In previous studies (Juang and Lee, 1998a; Juang et al., 1999), we found that the spatial distributions of heavy metals in contaminated soils have great variation and high skewness. Moreover, there are a few cases where locally extreme values are surrounded by much smaller values. In this situation, there will be huge spatial variation among observations over a short distance, and the fitted semivariogram model usually has a large nugget effect. The large nugget effect means the variable is not very regular and is discontinuous. If a pure nugget effect happens, which entails a complete lack of spatial correlation, the kriging estimate will become a simple arithmetic average of sampled data and any map generated by using the kriging process will not be very meaningful.

Logarithmic transformation is one approach usually used to detect spatially dependent structures for highly skewed data (Cambardella et al., 1994; Litaor, 1995; Van Meirvenne et al., 1996). Logarithmic transformation is used for the data following a lognormal distribution and then the lognormal kriging estimator can be used for spatial interpolation (Journel, 1980). The lognormal kriging estimator provides an approximately unbiased estimate, although error estimations are often exaggerated. The lognormal kriging estimator only works well when the transformed data are a Gaussian random function. For highly skewed data, it is necessary to check whether the univariate distribution of data is lognormal or not before using lognormal kriging.

On the other hand, the normal-scored transformation is an alternative for dealing with the data, which have a positively skewed distribution with a few extreme values. This method can transform any data set having an asymmetric distribution into the normal scores, which have a standard normal distribution. Then, the kriging estimation can be performed in the normal-scored space. This approach is based on a multi-Gaussian model. Only when normal-scored data strictly follow the multi-Gaussian distribution (also called multi-normal distribution), the kriging estimation in the normal-scored space is valid. In practice, it is difficult to ensure that normal-scored data are multi-Gaussian. Goovaerts (1997) suggested that one should check whether normal-scored data are reasonably bi-Gaussian. If they are, then the multi-Gaussian model may be tenable; if they are not, another approach should be considered.

Both logarithmic and normal-scored transformations require that the transformed data should follow a specific distribution function (such as Gaussian or multi-Gaussian distribution). Unfortunately, real data sets may not meet with the severe requirements for using logarithmic and normal-scored transformations. Recently, Journel and Deutsch (1997) proposed an approach, termed rank-order geostatistics, for integration of information of diverse data types, scales, support, and accuracy. In this approach, the standardized rank transformation is used. There is not any specific requirement regarding the distribution of the transformed data. The kriging estimation can be performed in the standardized-rank space and then the kriging estimates can be back-transformed into the original space. Therefore, rank-order geostatistics is an alternative for dealing with highly skewed data. In addition, the standardized ranks follow a uniform distribution. One can determine the uniform distribution for any unsampled location based on the kriging estimate and variance. The uniform distribution can be used to calculate the probability of the attributes of interest being higher than a given cutoff value. For delineating hazardous areas of contaminated soils, this probability can be used to assess the delineation uncertainty and to provide a quantitative means with more information for determining whether cleanup actions are necessary.

To our knowledge, there was not any case study to use rank-order geostatistics for interpolation of highly skewed data in heavy-metal contaminated soils. The objective of this study was to demonstrate how rank-order geostatistics could be used to handle a data set of soil heavy-metal concentrations with large variation, high skewness, and some extreme values. A real data set of Cd concentrations in a contaminated site in Taiwan was presented for illustrating the utility and success of rank-order geostatistics with standardized rank transformation for the spatial interpolation when commonly used nonlinear methods, such as logarithmic and normal-scored transformations, are not suitable.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Soil Sampling and Heavy-Metal Measurements
The study site is a rice (Oryza sativa L.) paddy field about 5 ha in area situated in Taoyuan County, Taiwan. The paddy field was irrigated with water, which was contaminated with heavy metals in the discharge from a chemical plant, through irrigation channels. The chemical plant was located on the south side of the paddy field. The soil was contaminated by heavy metals in the discharge of the chemical plant. According to the U.S. soil taxonomy, the soil is a Typic Paleudult. Some characteristics of this soil are shown in Table 1. A systematic sampling scheme was used. Seventy-eight soil samples were taken on a 25- x 25-m grid covering the 5-ha site as shown in Fig. 1. The soil was collected with an auger to a depth of 15 cm. Three soil cores (inner diameter of auger = 7.5 cm) were bulked together into one composite sample for each sampling location. Soil samples were air-dried, ground, and sieved (<0.84 mm) prior to analysis. Ten grams of soil and 100 mL 0.1 M HCl were placed in a 250-mL flask. Each flask was shaken for 1 h, and then the soil suspensions were filtered through Whatman No. 42 filter paper. Cadmium in the filtrates was then determined by means of atomic absorption spectroscopy (Chen, 1991). There were three replicates on the heavy-metal measurements for each soil sample. The detection limit of 0.1 M HCl extractable soil Cd is 0.30 mg/kg.


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  		Table 1. Soil characteristics.

 


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  		Fig. 1. The study site and the sampling points. The coordinate system is 2°UTM for Taiwan.

 
Exploratory Data Analysis
The exploratory data analysis (EDA), including the post plot, summary statistics, histograms, Kolmogorov–Smirnov (K–S) test, semivariograms, and bivariate Gaussian check, was performed for the selected data set of soil Cd concentrations. Logarithmic and normal-scored transformations, y(x) and v(x), were used:

[1]
where z(x) is the soil Cd concentration, and:

[2]
where k is the rank of z(x) by ascending sorting n points of the data set and G is the standard Gaussian cumulative distribution function (CDF). Histograms and Kolmogorov–Smirnov (K–S) tests were used to check whether the data of soil Cd concentrations follow a lognormal distribution and to determine the feasibility of lognormal kriging. Semivariograms and a bivariate Gaussian check were used to assess whether the kriging estimation in the normal-scored space is valid. The bivariate Gaussian check is based on the comparison of experimental and Gaussian-based indicator semivariograms, {gamma}eI(h;vp) and {gamma}gI(h;vp):

[3]
where I(xi;vp) is the indicator at location xi transformed by the p quantile value vp in the normal-scored space, and:

[4]
where {gamma}v(h) is the semivariogram model of normal scores. If {gamma}eI(h;vp) is approximately equal to {gamma}gI(h;vp), the selected data will be consistent with a bivariate Gaussian distribution. Experimental and Gaussian-based indicator semivariograms were obtained using the BIGAUS program in the GSLIB library (Deutsch and Journel, 1997).

Rank-Order Geostatistics
Standardized Rank Transformation
Assume a continuous variable Z with a cumulative distribution function (CDF), F(z), and then F(Z) has a uniform distribution on the interval from zero to one. For a random sample, zi, i = 1, 2, ... , n, of Z, the empirical distribution function (EDF) Fe(z) can be used to estimate F(z). Arrange zi in ascending order and denote the order statistics by z(1) <= z(2) <= ... <= z(r) <= ... <= z(n), where z(r) is called the rth order statistic. The EDF, Fe[z(r)], for the rth order statistic then can be defined by:

[5]

The standardized rank transformation (U) is defined as:

[6]

Then, the distribution of U is also uniform. There is an interesting interpretation of the CDF, F(u), of U:

[7]

Suppose z(x) is the soil Cd concentration at location x. Consider a random sample with n data values z(xi), i = 1, 2, ... , n, and their rank orders r(xi). Based on Eq. [5] and [6], the standardized ranks u(xi) of the sample can be calculated by:

[8]

The value of u(xi) is between 1/n and 1. The standardized rank transformation is monotonously increasing. The u(xi) is thus always conditional on a random sample z(xi) of a given size n from the continuous variable Z(x). Therefore, the n data points of u(xi) can be regarded as a conditional realization of the spatial random field U(x) given the random sample z(xi).

Kriging of Standardized Ranks
For kriging estimation, the spatial dependence of standardized ranks u(x) should be quantified using the semivariogram:

[9]
where {gamma}U(h) is the semivariance, which is assumed to be isotropic with respect to different directions on a horizontal plain, h is the distance between two locations x and x + h, and N(h) is the number of pairs for x and x + h. Under the semivariogram model defined by Eq. [9], the standardized rank u(xo) at an unsampled location xo can be estimated by means of kriging. The ordinary kriging estimate, u*(xo), is used:

[10]
where the weights {lambda}i should sum to unity to ensure that u*(xo) is unbiased and m is the number of the selected surrounding observations used in the estimation. By minimizing the kriging variance, the m weights, {lambda}i, and one Lagrange multiplier, µ, can be obtained from the following system of m + 1 linear equations:

[11]
and

[12]

The kriging variance is defined by:

[13]

In addition, note the interpretation of Eq. [7]. The kriging estimate u*(xo) also means the probability Prob[U(xo) < u*(xo)]. Thus, u*(xo) should be strictly between 0 and 1.

Based on the kriging estimation, an estimated distribution for the variable U(xo) can be determined. The estimated distribution of U(xo) is a uniform distribution U(a,b) with a lower bound a and an upper bound b. Thus, the mean value µ(xo) and variance {sigma}2(xo) of U(xo) are respectively equal to (a + b)/2 and (b - a)2/12. If the kriging estimate u*(xo) and kriging variance sk2(xo) are regarded as the estimates of µ(xo) and {sigma}2(xo), the estimated distribution of U(xo) can be written as:

[14]
where a = u* - sk and b = u* + sk.

Back-Transformation to the Concentration Space from Standardized Ranks
The estimated value of u*(xo) will be between two adjacent standardized ranks, u(xi) = r/n and u(xj) = (r + 1)/n. Based on the transformation model of Eq. [6], the relation between z(x) and u(x) is monotonously increasing. The estimated value z*(xo) in the concentration space corresponding to u*(xo) will be between both concentration values, z(xi) and z(xj), which are corresponding to u(xi) and u(xj), respectively. Thus, one can use a specified interpolation model to estimate the value of z*(xo). Several methods have been proposed for the interpolation within a standardized-rank interval of an EDF (Deutsch and Journel, 1997; Goovaerts, 1997). The determination of an interpolation model is usually based on past experience with the phenomena or external information. In this study, the middle point model, z*(xo) = [z(x(r)) + z(x(r+1))]/2 was used. Note that the associated estimation error for z*(xo) is nonlinearly dependent on the kriging estimation error in the standardized-rank space and thus the estimate confidence interval in the concentration space will in general contract or expand relative to that in the standardized rank space. A hypothetical example is shown in Fig. 2 with the distribution of the original data exhibiting large variation and high skewness. A similar deviation, d, in the standardized-rank space may be back-transformed into greatly different uncertainty intervals in the concentration space. Thus, the changes of standardized rank interval during back-transformation also should be taken into account for assessing the variation of z*(xo). The lower and upper bounds, zL(xo) and zU(xo), in the concentration space can be obtained from u* - sk and u* + sk using the middle point model. The interval [zL(xo), zU(xo)] indicates the variation of Z(xo).



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  		Fig. 2. A hypothetical empirical distribution function (EDF) for a sample data set with great variation and high skewness used to illustrate a deviation, d, in the standardized-rank space back-transformed into greatly contrasting uncertainty intervals {delta} or {Delta} in the concentration space.

 
The probability Prob[Z(xo) < zc] of soil Cd concentration at location xo being lower than a selected hazardous threshold zc also can be obtained. Based on the transformation model of Eq. [6]:

[15]
where Prob[U(xo) < uc] is the probability of the standardized rank at location xo being lower than a specified cutoff value uc = F(zc). Prob[Z(xo) < zc] then can be estimated by using the distribution of U(xo) presented in Eq. [14]. The estimated probability Prob*[Z(xo) < zc] is shown as follows:

[16]
where < uc < . If uc is smaller than the lower bound, u* - sk, the probability Prob*[U(xo) < uc] is equal to 0. If uc is greater than the upper bound, u* + sk, Prob* should be equal to 1. The estimated probability of a concentration being higher than the cutoff value, Prob*[Z(xo) > zc], would straightly indicate a possibility of contamination. This probability can be derived from Eq. [16]:

[17]

Procedures
(i) First, the standardized rank transformation was carried out using Eq. [8]. If k observations are equal and are tied for ranks r, r + 1, ... , and r + k - 1, then each should be assigned the average rank, [r + (r + 1) + ... + (r + k - 1)]/k. The variogram model of standardized ranks was developed for kriging estimation. The experimental semivariogram and the fitted model were obtained using the GS+ software (Gamma Design, 1994).

(ii) The ordinary kriging estimator was used to estimate the standardized ranks at unsampled locations. The radius of the search area was limited to the range of the semivariogram model. The number of surrounding data adopted for each estimated value was set between 4 and 8 to avoid the occurrence of negative kriging weights. In practice, negative kriging weights may occur and the kriging estimated value may be out of the interval [0,1]. This problem can be resolved by using the ordinary kriging weights to build a posterior distribution function (Rao and Journel, 1997). One also can adjust the search area for kriging estimation to avoid unreliable estimated values. Thus, before adopting the final search parameters, we tried several search strategies over the entire area to make sure that the estimate, u*(xo), is strictly between 0 and 1. Then, the kriging estimated values u*(xo) and kriging variances s2k at 243 grid nodes were obtained. The GEO–EAS program (Englund and Sparks, 1988) was used to perform the ordinary kriging estimation.

(iii) To evaluate the reliability of kriging estimation in the standardized-rank space, cross-validation was used, with the mean error (ME) and mean square relative error (MSRE) of estimated values being calculated (Voltz and Webster, 1990). The ME is a measure of the bias of the estimation, and it should be close to zero for unbiased methods. It is defined by:

[18]
where u(xi) is the standardized rank transformed from the observed Cd concentration, z(xi), at each of 78 sampling points and u*(xi) is the kriging estimate of the standardized rank. The index MSRE is a measure of consistency for kriging estimation. The MSRE can be defined by:

[19]
where sk(xi) is the kriging standard deviation corresponding to u*(xi). A reliable estimation should have an MSRE value close to 1.

(iv) The estimated Cd concentration z*(xo) was obtained from back-transformation of the kriging estimate u*(xo) in the standardized rank space using the middle point model. The estimated probability Prob*[Z(xo) > 10] for soil Cd concentration at location xo being higher than the hazardous level threshold value zc = 10 mg/kg was also obtained by using Eq. [16] and [17]. The lower and upper bounds, u* - sk and u* + sk, of the uniform distributions for U(xo) were back-transformed into the concentration space to obtain the lower and upper bounds, zL(xo) and zU(xo), in the concentration space. The deviation between zL(xo) and zU(xo) was then used to indicate the variation of Z(xo).

(v) Moreover, in order to assess the reliability of estimated Cd concentrations back-transformed from the standardized-rank space, the kriging estimate u*(xi) and the kriging standard deviation sk(xi) obtained form the cross-validation were used. The concentration interval [zL(xi), zU(xi)] was obtained from for each of 78 sampling points. An index, R, as shown in the following was used to evaluate the reliability of back-transformation:

[20]
where I(xi) is 1or 0. If the observed value of z(xi) is located in the concentration interval, [zL(xi), zU(xi)], then I(xi) = 1; otherwise, I(xi) = 0. The higher the index R, the more accurate the estimation of the soil Cd concentrations. The variance of R also can be estimated by using R(1 - R)/78 to evaluate the precision of R.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Preliminary Data Description by Exploratory Data Analysis
Table 2 presents descriptive statistics for the soil Cd concentrations. The large coefficient of variation (CV) indicates large variability of soil Cd at this site. The high skewness suggests the presence of extreme values in the data. On the post plot in Fig. 3, several local extreme values, which are surrounded by smaller values, could be found. The Kolmogorov–Smirnov (K–S) test on both Cd and log(Cd) shown in Table 2 indicated that the observed distribution of soil Cd concentrations was significantly different from either a normal or a lognormal distribution at the level of 0.05. The histograms of soil Cd concentrations and logarithmic-transformed values are shown in Fig. 4(a) and 4(b), respectively. It clearly reveals that the data of soil Cd concentrations are highly skewed and that the logarithmic-transformed data diverge from a normal distribution. Thus, the kriging estimation in the logarithmic space should be performed with caution because the lognormal kriging estimation is nonrobust against departures from the lognormal distribution (Chilès and Delfiner, 1999). In fact, the problem lies in the back-transform. The back-transform through exponentiation tends to exaggerate any error associated with the kriging estimation (Goovaerts, 1997):

[21]
where y* and s2lk are the kriging estimated value and kriging variance in the logarithmic space, respectively. Moreover, the lognormal kriging estimate (z*lk) is conditional unbiased and hinges on the assumption of small kriging variance and the correct determination of the variogram sill. The estimate z*lk is very sensitive to variogram fluctuations (Chilès and Delfiner, 1999). Therefore, the logarithmic transformation is not suitable for the data of soil Cd concentrations.


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  		Table 2. The descriptive statistics of the soil Cd concentrations in the study site (n = 78).

 


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  		Fig. 3. Sampling location map and soil Cd concentrations.

 


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  		Fig. 4. Histograms of (a) soil Cd concentrations and (b) logarithmic-transformed values.

 
Figure 5 shows the semivariograms of soil Cd concentrations and normal-score transformed values. The contrast between the semivariograms of soil Cd concentrations and normal-score transformed values is clearly evident. The experimental variogram (Fig. 5a) of soil Cd concentrations indicates minimal spatial dependence and it was difficult to fit a reliable model for the kriging estimation, but the normal-scored variogram (Fig. 5b) exhibits more obvious spatial dependence. The normal-scored transformed values are distributed symmetrically without high skewness or extreme variation. However, the feasibility of the kriging estimation in the normal-scored space is dependent on the conformity with the multivariate Gaussian model, as indicated by the results of the bivariate Gaussian check (Deutsch and Journel, 1997). Figure 6 shows the experimental and Gaussian-based indicator variograms for bivariate Gaussian comparison. The experimental indicator variograms depart significantly from the Gaussian-based indicator variograms. This suggests that the normal-scored data do not follow a bivariate Gaussian distribution (Goovaerts, 1997). Thus, the kriging estimation in the normal-scored space is suspect. This implies that it is unsuitable to use the normal-scored transformation for spatial interpolation of the soil Cd concentrations. At present, there is not a valid method to directly ensure the normal scores of the real data are multi-Gaussian. Journel and Deutsch (1997) suggested that the normal-scored transformation is useful only if one wishes to call on the congenial but restrictive properties of the multi-Gaussian model.



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  		Fig. 5. Semivariograms of (a) soil Cd concentration values and (b) normal-scored values.

 


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  		Fig. 6. Experimental indicator semivariograms for the (a) 5th, (b) 25th, (c) 50th, (d) 75th, and (e) 95th percentiles of the sample distribution. The solid lines depict the indicator semivariograms for the multi-Gaussian distribution.

 
Mapping Soil Cadmium Concentrations Using Rank-Order Geostatistics
The results of exploratory data analysis (EDA) indicate that the logarithmic and normal-scored transformations are unsuitable for spatial interpolation of the soil Cd concentrations. Thus, rank-order geostatistics was adopted to perform the spatial interpolation of soil Cd concentrations. The empirical distribution function (EDF) of soil Cd concentrations is shown in Fig. 7. The standardized rank means the relative cumulative frequency. Most of the soil Cd concentrations are lower than 20 mg/kg whereas the maximum value is almost 80 mg/kg. This figure also illustrates the large skewness in the original data, as well as the relatively smooth transition from the original data to standardized ranks. Moreover, Fig. 8 shows the spatial structure of standardized ranks. A spherical model with a sill of 0.097, nugget of 0.022, and range of approximately 117 m was developed for the variogram. The spatial structure evident in the standardized-rank variogram suggests the possibility of meaningful kriging estimation. Based on the variogram model of standardized ranks, the kriging estimation was carried out in the standardized rank space. The results of cross-validation showed that the mean error (ME) of the estimated standardized ranks is less than 0.01 and the mean square relative error (MSRE) of the estimated standardized ranks is almost equal to 1.00. This suggests that the kriging estimation in the standardized rank space is reliable.



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  		Fig. 7. Empirical distribution function for the soil Cd concentrations.

 


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  		Fig. 8. Semivariogram of standardized ranks for Cd concentrations.

 
The kriging estimates of standardized ranks were back-transformed into the Cd concentrations based on the EDF curve in Fig. 7. Figure 9 shows the contour map of the estimated soil Cd concentration. It is evident that there are some highly contaminated regions, so-called hot spots, on the map. The areas surrounding the hot spots have large concentration gradients. Figures 10(a) and 10(b) show the estimated lower and upper bounds of the concentration interval for the soil Cd concentrations. The contour map of the interval range (the deviation between the upper and lower bounds) of estimated soil Cd concentration is shown in Fig. 11(a), which indicates the variation of soil Cd concentration after back-transformation from standardized ranks. The interval ranges in some areas are very large, as a result of the large variation in the observed Cd concentrations. The large interval ranges are all adjacent to the hot spots. This suggests that the areas close to the hot spots have large uncertainty in the soil Cd concentrations. Moreover, in order to show the influence of back-transformation from the standardized rank on the variation of kriging estimation for the soil Cd concentration, the contour map of kriging standard deviations of standardized ranks is also shown in Fig. 11(b). Since the systematic sampling scheme was used in the study and the kriging standard deviation of the standardized rank is dependent upon the configuration of sampling points, the variation of standardized ranks distributed homogeneously. Comparing Fig. 11(a) with 11(b), the spatial variation in the Cd concentration space after back-transformation is obviously different from that in the standardized-rank space. Thus, one can find that kriging of the nonlinear standardized rank transform provides an estimation variation that is a function of not only the data configuration, but the data values also. The index R, which indicates the ratio of observed soil Cd concentrations located in the estimated concentration intervals of [zL(xi), zU(xi)] from the cross-validation results, is 0.92. The 95% confidence interval of R is [0.86, 0.98]. The high ratio of R demonstrates that the spatial estimates of soil Cd in the concentration space, which were back-transformed from estimated standardized ranks, are reliable.



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  		Fig. 9. Contour map of estimated Cd concentrations.

 


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  		Fig. 10. Contour maps of the (a) lower and (b) upper bounds of the concentration interval for soil Cd concentrations.

 


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  		Fig. 11. Contour maps of (a) the interval range (the deviation between the upper and lower bounds) for soil Cd concentrations and (b) the kriging standard deviation for standardized ranks.

 
Delineation of Hazardous Regions
The probability of a pollutant being higher than a cutoff value also can be obtained using the rank-order geostatistical approach. In a heavy-metal contaminated site, the probability of heavy-metal concentrations being higher than a cutoff value can be used to indicate the possibility of contamination. A decision-maker can straightly use this probability assessment to determine whether or not some unsampled locations or areas require cleanup (Juang and Lee, 1998b; Juang and Lee, 2000). The probability of a Cd concentration higher than 10 mg/kg is shown in Fig. 12. One can see that the areas with a large probability of Cd concentration higher than 10 mg/kg coincide with the areas of high Cd concentration in Fig. 9 and 10. For delineation of hazardous areas based on a cutoff value, the contour maps of the upper and lower bounds of the estimated pollutant concentration and the associated probability of being higher than a cutoff value can be used. If both the lower and upper bounds of the estimated pollutant are larger than the cutoff value at unsampled locations, then these locations can be delineated as hazardous. On the other hand, if both the lower and upper bounds of estimated pollutant are smaller than the cutoff value, then these locations can be delineated as nonhazardous. If the concentration interval of an unsampled location includes the cutoff value, then delineation is more challenging. In this situation, the probability of being higher than a given cutoff value provides quantitative information about the uncertainty of delineation.



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  		Fig. 12. Contour map of the probability of soil Cd concentration z(x) being higher than 10 mg/kg.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
This paper showed that rank-order geostatistics could be used to handle a data set of soil heavy-metal concentrations with large variation, high skewness, and some extreme values. When commonly used approaches, such as logarithmic and normal-scored transformations, are not suitable for the data set, the standardized rank transformation provides a possible alternative. The spatial structure of the standardized rank transformation can be used as an alternative to attempting to identify the spatial dependence of the observed values, which is often masked by the high skewness and large variation of the original observations. The kriging estimation was performed smoothly in the standardized rank space. The kriged ranks were also back-transformed into the original space. Thus, the spatial distributions of the heavy-metal concentrations and the probabilities of heavy-metal concentration being higher than a given threshold were reliably estimated using the rank-order geostatistical approach.


   ACKNOWLEDGMENTS
 
This research was partly sponsored by the National Science Council, Republic of China under Grant no. NSC-87-2621-P-002-013 and NSC 89-2621-B-002-007. Kai-Wei Juang and Dar-Yuan Lee thank the National Science Council, Taiwan, Republic of China for financial support for their visit to the University of Illinois at Urbana–Champaign, where they completed this manuscript.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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J.A. Chipres, J.C. Salinas, J. Castro-Larragoitia, and M.G. Monroy
Geochemical mapping of major and trace elements in soils from the Altiplano Potosino, Mexico: a multi-scale comparison
Geochemistry: Exploration, Environment, Analysis, November 1, 2008; 8(3-4): 279 - 290.
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The SCI Journals Agronomy Journal Crop Science
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