Spatial interpolation of severely skewed data with ... - Springer Link

Environ Earth Sci (2011) 63:1093–1103 DOI 10.1007/s12665-010-0784-z

ORIGINAL ARTICLE

Spatial interpolation of severely skewed data with several peak values by the approach integrating kriging and triangular irregular network interpolation Chunfa Wu • Jiaping Wu • Yongming Luo • Haibo Zhang • Ying Teng • Stephen D. DeGloria

Received: 27 October 2009 / Accepted: 1 October 2010 / Published online: 19 October 2010  Springer-Verlag 2010

Abstract It was not unusual in soil and environmental studies that the distribution of data is severely skewed with several high peak values, which causes the difficulty for Kriging with data transformation to make a satisfied prediction. This paper tested an approach that integrates kriging and triangular irregular network interpolation to make predictions. A data set consisting of total Copper (Cu) concentrations of 147 soil samples, with a skewness of 4.64 and several high peak values, from a copper smelting contaminated site in Zhejiang Province, China. The original data were partitioned into two parts. One represented the holistic spatial variability, followed by lognormal distribution, and then was interpolated by lognormal ordinary kriging. The other assumed to show the local variability of the area that near to high peak values, and triangular irregular network interpolation was applied. These two predictions were integrated into one map. This map was assessed by comparing with rank-order ordinary kriging and normal score ordinary kriging using another data set consisting of 54 soil samples of Cu in the same region. According to the mean error and root mean square error, the approach integrating lognormal ordinary kriging and triangular irregular network interpolation could make C. Wu  Y. Luo (&)  H. Zhang  Y. Teng Key Laboratory of Soil Environment and Pollution Remediation, Institute of Soil Science, Chinese Academy of Sciences, Nanjing 210008, China e-mail: [email protected] C. Wu  J. Wu College of Environmental and Resources Sciences, Zhejiang University, Hangzhou 310029, China S. D. DeGloria Department of Crop and Soil Sciences, Cornell University, Ithaca, NY 14853, USA

improved predictions over rank-order ordinary kriging and normal score ordinary kriging for the severely skewed data with several high peak values. Keywords Skewed data  High peak value  Data transformation  Triangular irregular network  Spatial distribution Abbreviations Cu TIN OK OKRK OKNS OKLG ? TIN

LG RK RS IDW Cdf ME RMSE COV

Copper Triangular irregular network Ordinary kriging Rank order ordinary kriging Normal score ordinary kriging Integration of lognormal ordinary kriging and triangular irregular network interpolation Logarithmic transform Rank order transformation Normal score transformation Inverse distance weighted Cumulative distribution function Mean error Root mean square error Coefficient of variation

Introduction Soil contamination by heavy metals is serious in China (Luo et al. 2003; Li 2005; Tan et al. 2006) and some area of other countries (Bardgett et al. 1994; Loland and Singh 2004). Copper (Cu) is one of the heavy metals most often

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encountered (Cao and Hu 2000; Lu et al. 2003). Cu contamination not only directly affects soil physical and chemical properties, and decreases nutrient availability (Moreno et al. 1997), but also poses a threat to human health by entering food chains. In order to develop effective management practices for soil contamination, it is necessary to know the spatial distribution of pollutants. Geostatistical methods can provide an unbiased prediction with minimum variance for the content of a given pollutant (Isaaks and Srivastava 1989; Goovaerts 1997). Many studies used geostatistics to describe the spatial distribution of pollutants in contaminated soils (Atteia et al. 1994; Arrouays et al. 1996; Goovaerts 1997; Meuli et al. 1998; Carlon et al. 2001) and assess risk of exceeding critical threshold value (von Steiger et al. 1996; Baraba´s et al. 2001; van Meirvenne and Goovaerts 2001; Cattle et al. 2002). Geostatistical inferences using kriging techniques are more efficient when data for variables are distributed normally. However, previous studies showed that the spatial distributions of pollutants in contaminated soils varied greatly with a large skewness (Juang and Lee 1998; Juang et al. 1999). High peak values (i.e., locally extreme values are surrounded by much smaller values) were often observed (Hendficks Franssen et al. 1997). Difficulties caused by severely skewed distributions can often be alleviated by appropriate transformation of the data. The most common is the natural logarithmic transform (LG) (Journel 1980; Saito and Goovaerts 2000), which is best suited to lognormal data. A second approach is to use a standardized rank order transformation (RK) prior to kriging, a simple method that is well suited to integrating many diverse types of data (Journel and Deutsch 1997). A third approach is normal score transformation (NS) (Goovaerts 1997; Deutsch and Journel 1998), a procedure that transforms any distribution into a normalized distribution. To severely skewed data with several high peak values, data transformations may help correct the skewness, but not able to alleviate huge smooth effect on high peak values inherited from interpolation procedures such as kriging (Wu et al. 2006). These high peak values are usually surrounded by much smaller values. Smooth effect induced by interpolation leads to under-estimate large values and over-estimate small values, thus large errors occur around high peak values. Although the number of high peak values is limited, they are very important for pollution sources detection and pollution approach survey (Lee and Juang 2003). In general, the high peak values are simply deleted or replaced as outliers in conventional kriging for soil prosperities (Liu et al. 2004; Zhang and McGrath 2004; Rawlins et al. 2005; Zhang et al. 2006). This action will alleviate the smooth effect in kriging, but it is unacceptable for it conceal the related information of pollution sources in environmental study. Spatial point-

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process analysis (SPPA) is a new method that can alleviate the smooth effect on high peak values (Walter et al. 2005). However, this method was not useful for sparsely sampled, and it was difficult to use it for common environmental and soil research. Geostatistical methods quantify the spatial structure of the contaminant and then make prediction. Triangular irregular network (TIN) interpolation is one of conventional methods that predict unknown point using the values of three closest corner points of a triangle. It can eliminate the smooth effect of high peak values in neighboring area in interpolation, since the impact scope of individual node is strictly limited in an area between adjacent nodes. Therefore, integrating geostatistics and TIN method has the potential to share their merits and satisfy the need of spatial interpolation for severely skewed data with several high peak values. The objectives of this study is to evaluate the spatial distribution of severely skewed data with several high peak values predicted by integrating lognormal ordinary kriging and TIN interpolation, and to compare the result with those obtained by rank-order ordinary kriging and normal score ordinary kriging, respectively.

Materials and methods Descriptions of study area The study area is located in Fuyang valley, southwest of Hangzhou City, Zhejiang Province, China with a total land area of 10.9 km2 (Fig. 1). Its topography is characterized by hilly mountains in the periphery and plain in the center. The dominant wind directions are north and northeast. Many small copper smelting plants were found in study area in 1980s, which is assumed to be the major source of heavy metals contaminations in this small area. In recent years, most smelting plants had been closed for environmental reasons, and eleven smelters are still operating (the locations are shown in Fig. 4). Sample collection and analyses A total of 147 surface (0–20 cm) soil samples for prediction was collected in winter 2003 using a mixed grid sampling method. In spring 2004, 54 supplementary soil samples for validation were collected (Fig. 1). The samples for validation were collected on approximate ‘‘S’’ shaped (Huang et al. 2006) sampling method with consideration of the locations of smelting plants and the variability of soil Cu concentration. The total concentration of Cu was analyzed using atomic absorption spectroscopy (AAS) (Varian 220AS) after the samples was digested with aqua regia. The correctness of methodology was checked on the basis

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the extreme potential outliers that can distort the computing of a measure of spread and lessen the sensitivity to outliers. In the terminology of Tukey (1977), the fences procedure uses the estimated interquartile range (IQR), referred to as the Hspread, which is the difference between values of the hinges, i.e. sample third and first quartiles. Specifically, the inner fences, f1 and f3, and outer fences, F1 and F3, are usually defined as: f1 ¼ q1  1:5Hspread f3 ¼ q3 þ 1:5Hspread F1 ¼ q1  3:0Hspread F3 ¼ q3 þ 3:0Hspread

Fig. 1 Locations of sampling sites and study area

of reference materials IEAE-Soil-7 and CRM 277 (Loska et al. 2005). High peak value identification and data split Outliers are observations presumably come from a different distribution than the majority of the data. They can have a profound influence on the data analysis, often leading to erroneous conclusions (Schwertman and de Silva 2007). Outliers include extremely large values and small values. In environmental study, only extremely large values i.e. high peak values are flagged, but not extremely small ones. The large values produced by pollution are what people concern and have major influence on statistics and spatial prediction. Due to the major influence of outliers on most parametric tests, considerable attention has been devoted to the detection and accommodation of outliers (i.e. Andrews 1974; Tukey 1977; Atkinson 1994; Schwertman et al. 2004, Das and Basudhar 2006). Tukey (1977) suggested a simple graphical method for identifying outliers that is based on the box-plot and involves the construction of ‘‘inner fences’’ and ‘‘outer fences’’. This method is especially appealing not only in its simplicity but, more importantly, because it does not use

where q1 and q3 are the first and third sample quartiles, and Hspread = q3 - q1. Tukey (1977) referred to observations that fall between the inner and outer fences as ‘‘outside’’ outliers, while those that fall below the outer fence F1 or above the outer fence F3 are ‘‘far out’’ outliers. In this study, we used ‘‘outer fences’’ i.e. F3 to identify the high peak values. When the high peak values were identified from original data, we split all the original data for prediction into two parts,i.e. Part A and Part B. High peak values, as one kind of outliers, are usually presumed to come from a different distribution than the majority of the data. Part A reflects the holistic spatial variability, and Part B, the local spatial variability of the region that nears high peak values. To common values, Part A equals itself and Part B is zero. To high peak values, Part A equals the median of all samples except for the high peak values and Part B equals to itself minus Part A. Therefore, we divided the original data of all soil samples for prediction into two new data sets: set A and set B. The data set A and data set B made up of Part A and Part B of all soil samples for prediction, respectively. Data transformation and interpolation methods The rank order transformation and back-transformation are carried out as follows (Journel and Deutsch 1997; Juang et al. 2001): 1.

Arrange the n sample in ascending order: zð1Þ      zðrÞ      zðnÞ

ð1Þ

where the superscript r is the rank of datum z(r) among all n data, z(r) is the rth order statistic. 2. Calculate the standardized rank y(r) of the sample r ð2Þ yðrÞ ¼ n The value of y(r) is between 1/n and 1. 3. Rank order ordinary kriging (OKRK) is performed on standardized rank order transformed data. Estimated

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ranks, y ðuÞ, are back-transformed into the original units for variable Z: z ðuÞ ¼ F 1 ðy ðuÞÞ

ð3Þ

-1

F () is the inverse function of the function y (). Most estimated values for y ðuÞ usually fall between two adjacent standardized ranks, say r/n and (r ? 1)/n. Under the circumstances, the corresponding estimates in the original concentration space z ðuÞ will be between z(r) and z(r?1). Thus, the value of z ðuÞ is assigned to the mid-point between z(r) and z(r?1) (Juang et al. 2001): h i z ðuÞ ¼ 0:5 zðrÞ þ zðrþ1Þ ð4Þ If y ðuÞ happens to be r/n, then z  ð uÞ ¼ z ð r Þ

ð5Þ

On occasion, a value for y ðuÞ estimated by kriging may fall outside the acceptable range between the minimum of 1/n and the maximum of 1. In this case, we re-assigned any estimate \1/n to equal 1/n and any estimate [1 to equal 1, prior to back-transformation. The normal score transform is carried out as follows (Deutsch and Journel 1998; Saito and Goovaerts 2000): 1.

The n sample data are ranked in descending order: zðnÞ      zðkÞ      zð1Þ

ð6Þ (k)

where the superscript k is the rank of datum z among all n data; 2. The sample cumulative frequency of the datum z(k) is then computed as: pk ¼ k=n

ð7Þ

3.

The normal score transform of the z(k) datum is matched to the pk quantile of the standard normal cumulative distribution function (cdf): n o   ð8Þ yðkÞ ¼ G1 F½zðkÞ  ¼ G1 pk

4.

Normal score ordinary kriging (OKNS) is performed on the normal score transformed data. Estimates of the standard normal deviate, y ðuÞ, are back-transformed to original unit: z  ðuÞ ¼ F 1 ðGðy ðuÞÞÞ

ð9Þ

where F(z) is the cumulative distribution function (cdf) of the original data. Lognormal ordinary kriging is performed on natural logtransformed data. Back-transformation of kriging result was carried out by exponentiation (exp (z)), providing a prediction for total Cu (Cu) concentration expressed in original concentration units.

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TIN interpolation includes three steps. The first step is triangulation that takes the grid points and triangulates them using the Delaunay triangulation method (Park et al. 2001). The next step is determines which triangle the sample point lies within. The last step is linear interpolation. The linear interpolation is done by calculating the plane equation that fits through the three grid points at the triangle vertices, then solving for the value using the coordinates of the sample point and the plane equation. Based on three corner points, a planar plane can be fitted to each triangle using the equation: Z ¼ a X þ b  Y þ c

ð10Þ

where a, b and c are parameters, X and Y is X-coordinate and Y-coordinate of location, respectively, and Z is the predicted variable. The value of a, b and c can be solved by use the three corner points, then variable value for any unknown location that lies within the triangle can be estimated by the equation (Eq. 10) to interpolate. Inverse distance weighted (IDW) interpolation is one of the most commonly used techniques for interpolation of scatter points (Mueller et al. 2004). IDW is based on the assumption that the interpolating surface should be influenced most by the nearby points and less by the more distant points. The interpolating surface is a weighted average of the scatter points and the weight assigned to each scatter point diminishes as the distance from the interpolation point to the scatter point increases. LG, RK, and NS are the three most common data transformation methods for severely skewed data. In this study, OKRK and OKNS methods worked on original data of all soil samples for prediction before peak value removal; however, OKLG only worked on the data set A, and both IDW and TIN only worked on the data set B. All kriging inferences were made using GSLIB (Deutsch and Journel 1998), TIN interpolation, relative file conversion (TIN to grid file, grid to image), and predicted map integration were conducted using ArcGIS 9.0. The IDW interpolation was also conducted using ArcGIS 9.0, and the detail process can be found in ArcGIS Desktop Help 9.0. Evaluation of kriging methods Fifty-four soil samples that were sampled in spring 2004 as supplement were used for validation. To evaluate the performance of the three spatial methods, descriptive statistics were used to compare measured concentrations of 54 soil samples for validation with the predicted ones by the three interpolation methods. In addition, the mean error (ME), and root mean square error (RMSE) were calculated. The ME and RMSE have their standard meanings (Isaaks and Srivastava 1989):

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boundary of study area and outside the borders of corresponding prediction.

Results Status of soil Cu pollution and high peak values

Fig. 2 Histogram of copper concentrations of 147 soil samples in study area n 1X ½zðui Þ  z ðui Þ n i¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X RMSE ¼ ½ z ð ui Þ  z  ð ui Þ  2 n i¼1

ME ¼

ð11Þ

ð12Þ

where z(ui) is the measured value of Z at location ui, z ðui Þ is the predicted value at the same location. The ME provides a measure of bias; the RMSE provides a measure of accuracy. To consider of the difference of sampling program in winter 2003 and spring 2004, 20 soil samples were selected randomly from the 147 soil samples that were collected in winter 2003 for validation to evaluate the performance of three spatial methods on the other 127 soil samples for prediction, and repeated it 10 times. The average ME and average RMSE of 10 times were calculated for evaluation. In the ME and RMSE calculation, several samples for validation were excluded for they were located in the

The Cu concentrations of the 147 samples for prediction ranged from 4.2 to 3,065.1 mg kg-1 with a mean of 231.5 mg kg-1 (Fig. 2). The Cu concentrations of 54 samples for validation were in the range of 11.8–2,087.7 mg kg-1, and had a mean concentration of 292.0 mg kg-1. About 66% of soil samples for prediction (97 samples) and 74% of soil samples for validation (40 samples) had the Cu concentration exceeding the guide value (50 mg kg-1 for soil pH value \6.5 and 100 mg kg-1 for pH Cto 6.5) according to the Chinese Environmental Quality Standard for Soils (GB 15618-1995) (State Environmental Protection Administration of China 1995). The Cu concentrations of about 96% soil samples for prediction (141 soil samples) were in the range of 4.2–763.8 mg kg-1, and the Cu concentrations of the rest six samples were as high as 932.9, 1,070.5, 1,737.5, 1,764.8, 2,640.5 and 3,065.1 mg kg-1, respectively. They are belonging to high peak values according to ‘‘outer fences’’. The 54 samples for validation also had four very high peak values using the ‘‘outer fences’’ value of the 147 soil samples for prediction, and they were 961.3, 1,019.3, 1,750.7 and 2,087.4 mg kg-1, respectively. The original data of 147 soil samples that were collected in winter 2003 had a severely skewed distribution with a very large skewness (4.64) and also a very large COV (175.1%) (Table 1). They did not follow normal distribution or lognormal distribution according to the

Table 1 Summary statistics for the original data and transformed data of copper concentrations in soils (mg kg-1) Original data All

Transformed data Exclusive

Substitute

RK

NS

LG

Maximum

3,065.1

763.8

763.8

1.00

2.71

6.64

Minimum

4.2

4.2

4.2

0.00

-2.71

1.42

Mean

231.5

161.8

159.4

0.50

0.00

4.66

Median

106.5

102.1

102.1

0.50

0.00

4.63

SD

405.3

161.6

158.7

0.29

1.00

0.93

COV (%)

175.1

99.8

99.5

58.0

–

20.0

Skewness Kurtosis

4.64 25.08

1.79 2.69

1.85 2.98

0.00 -1.20

0.00 -0.12

-0.09 0.26

K–S test

Non-Normal

Non-Normal

Non-Normal

Normal

Normal

Normal

SD standard deviation, COV coefficients of variation, All all original data, Exclusive all original data except the six high peak values, Substitute the original data that high peak values were substituted by the median of the other, K–S test Kolmogorov–Smirnov test at the 0.05 level, RK rankorder, NS normal score, LG the natural logarithm of the original data that high peak values were substituted by the sample median

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Kolmogorov–Smirnov test. The skewness decreased sharply from 4.64 to 1.79 and the coefficient of variation (COV) also decreased from 175.1 to 99.8% when the six high peak values were excluded from the original data of 147 soil samples for prediction (Table 1). This indicated that the high skewness of the original data was in large part due to the six high peak values. Spatial pattern of Cu concentration All three computed semivariograms of rank order transformed, normal score-transformed and log-transformed data of using data set A were very similar (Fig. 3). They were fitted with a spherical model, and had moderate nugget/sill ratio. The nugget/sill ratio and range distance were 40.4% and 1,957 m for the rank order transformed data 42.3% and 1,989 m for the normal score transformed data, and 50.0%, and 1,905 m for the log-transformed data, respectively. All Nugget/Sill ratio values of three

Fig. 3 Selected experimental semivariograms and fitted parameters of spherical model for three kinds of kriging predictions

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semivariograms were in the range of 25–75%. The Nugget/ Sill ratio was assumed to be a criterion to classify the spatial dependence. The ratio values lower than 25% and higher than 75% corresponded to strong and weak spatial dependency, respectively, while the ratio values between 25 and 75% corresponded to moderate spatial dependence (Cambardella et al. 1994). Thus, Cu concentrations in this sampling scale had a moderate spatial dependency. The predicted maps from three spatial prediction methods (OKRK, OKNS and OKLG ? TIN) were general similar. The Cu concentration in the west was higher than that in the east, and the central was higher than the north and the south in three prediction maps. Comparison and validation of prediction methods Three predicted maps by the three interpolation methods showed similarity in general, but had substantial differences in the areas around the six high peak values (Fig. 4). By comparing the map of OKLG ? TIN, the maps of OKRK and OKNS had smoother gradient, and some high peak values and local variability were degraded or disappeared. To the non-peak value locations, the predictions by OKRK, OKNS and OKLG had no obvious difference except the predictions by OKLG were less than that by OKRK and OKNS in the neighbor area of high peak values. Summary statistics for Cu concentrations estimated by three methods for the 54 validation samples are tabulated in Table 2. For comparison, this table includes also the summary of the true Cu concentrations at these sites. From the Table 2, we can found that the OKLG ? TIN not only improve the precision of the whole prediction, but also improve the precision of non-peak locations. Scatterplots between measured values and estimates by the three methods for validation samples are shown in Fig. 5. To the majority sampling sites, the predicted values by OKLG ? TIN were closer to the measured values than that of by OKRK and OKNS. It is clear that the predictions of Cu concentrations were improved substantially by OKLG ? TIN. The ME and RSME of 20 randomly selected soil samples for validation by three prediction methods in 10 times are tabulated in Table 3. From the Table 3, we can found that both the average ME and average RMSE of the prediction by OKLG ? TIN were less than that by OKRK, OKNS, suggesting the prediction by OKLG ? TIN was better than that of by OKRK, OKNS again. The data set B was also predicted by IDW with power 5 distances (IDW5), power 10 distances, power 15 distances, power 20 distances and power 30 distances (IDW30). The predictions by the IDW with 5 powers were very close. In this paper, only the predicted map of IDW5 and IDW30 were showed in Fig. 6 for they are the most representative in all 5 predicted maps. To evaluate the performance of

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Fig. 4 The prediction maps of copper concentration in study area by rank order ordinary kriging (OKRK), normal score ordinary kriging (OKNS) and integration of lognormal ordinary kriging (OKLG) and triangular irregular network interpolation (TIN), respectively

Table 2 Summary statistics for measured and predicted Cu concentrations (mg kg-1) of 54 (50) soil samples for validation by three prediction methods Measured

Predicted OKRK

OKNS

OKLG ? TIN

OKaRK

OKaNS

OKLG ? TINa

840.0

694.8

1,056.0

2,210.0

582.0

755.0

625.1

11.8

48.3

41.5

41.1

48.3

41.5

41.1

292.0 154.5

199.0 150.1

202.0 138.9

223.8 144.3

298.4 174.6

169.1 138.2

181.6 136.4

184.7 163.7

SD

396.6

179.3

215.0

228.2

416.7

136.0

164.0

140.9

LQ

73.2

72.1

85.2

86.6

78.7

82.5

85.2

78.2

UQ

347.5

286.66

177.6

210.7

253.7

164.5

172.7

196.8

True

True

Max

2,087.7

Min

11.8

Mean Med

ME RMSE

a

90.0

68.2

5.1

27.7

16.1

13.2

274.7

229.1

89.9

103.2

87.0

83.9

Max maximum, Min minimum, Med median, SD standard deviation, LQ the lower quartile (mg kg-1), UQ the upper quartile (mg kg-1), ME mean errors, RMSE root mean square errors, OKRK rank-order ordinary kriging, OKNS normal score ordinary kriging, OKLG ? TIN combine lognormal ordinary kriging with triangular irregular network interpolation a

Statistical summary for non-peak locations (all samples for validation except the four high peak values)

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Environ Earth Sci (2011) 63:1093–1103 Table 3 The mean errors (ME) and root mean square errors (RSME) of 20 randomly selected soil samples for validation (mg kg-1) by three prediction methods in 10 times Seriesa

ME

RMSE

OKRK

OKNS

OKLG ? TIN

OKRK

OKNS

OKLG ? TIN

1 (1)

75.2

72.4

63.8

241.3

239.8

224.9

2 (2)

63.1

62.7

54.1

194.8

185.7

161.3

3 (1)

73.5

72.1

62.7

239.4

238.7

219.6

4 (0)

32.7

33.1

19.4

83.6

82.1

57.0

5 (0)

37.6

38.3

22.3

86.3

85.8

58.9

6 (0)

35.2

36.4

20.7

77.4

73.5

41.6

7 (0)

26.3

24.9

15.6

48.2

43.7

28.3

8 (1)

89.2

86.8

83.5

233.5

230.2

228.1

9 (1)

54.4

52.7

47.8

209.2

202.7

195.4

10 (1) Average

42.3 53.0

45.8 52.5

48.3 43.8

222.8 163.7

220.9 160.3

215.2 143.0

OKRK rank-order ordinary kriging, OKNS normal score ordinary kriging, OKLG ? TIN combine lognormal ordinary kriging with triangular irregular network interpolation, Average the average ME or RMSE of all 10 times a

1 (1), 2 (2),…,10 (1), the outside of parentheses is the series number and the inside of parentheses is the number of high peak values

data set B prediction. The result also showed that the ME and RMSE of OKLG ? TIN was lower than that of OKLG ? IDW (5) and OKLG ? IDW (30).

Discussion

Fig. 5 Measured copper (Cu-measured) concentrations of the 54 soil samples for validation and their predicted concentrations (Cupredicted) by rank order ordinary kriging (a), normal score ordinary kriging (b) and integration of lognormal ordinary kriging and triangular irregular network interpolation (c), respectively

OKLG ? IDW, the 54 soil samples that were sampled in spring 2004 were used to validation. The ME and RMSE was for OKLG ? IDW (5), 14.6, 92.5 mg kg-1; OKL-1 G ? IDW (30), 18.3, 112.4 mg kg , respectively, suggesting that the IDW (5) was better than IDW (30) for the

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In environmental study, high peak value as one of outlier is significant greater than the values of majority samples. The univariate statistics of Grubbs, Dixon, Walsh, t test, and range method (mean ± n 9 SD), are quite commonly used to detect outlying values (Zhang and Selinus 1998), and these method are also suit for high peak value identification. However, the results may have some difference for they based on different criterions in high peak values identification. The results of high peak value identification were also influenced the result of data split directly and total Cu prediction indirectly. Locations with high peak values (hot spots) were close to the smelting furnaces (Fig. 4). The hot spots were probably due to the atmospheric deposition and scrap material dumps from copper smelting. Some high peak values and local variability were degraded or disappeared in the prediction maps of OKRK and OKNS and the ranges of OKRK and OKNS were reduced in some degree, in large part because of the smooth effect inherent in the kriging and in part because of the effects of back-transformation.

Environ Earth Sci (2011) 63:1093–1103

Fig. 6 The prediction maps of soil copper concentration in study area based on the data set B by triangular irregular network interpolation (TIN), inverse distance weighted interpolation with power 5 distances (IDW5) and power 30 distances (IDW30), respectively

High peak values are always surrounded by comparatively lower values. So, the high peak value was decrease sharply and the values of surrounding area were increase in some degree for the smooth effect inherent in the kriging prediction. Due to OKLG ? TIN split data into two parts, and the main part of high peak values were predicted by TIN interpolation that could avoid smooth effect, this method could substantially alleviate smooth effect of kriging on high peak values in prediction although OKLG was still influenced by smooth effect of kriging The high peak values were substituted by the median of the remainder data in logarithm kriging. So, the prediction based on OKLG ? TIN was not influenced by high peak values. All summary statistics of the predictions by OKLG ? TIN except the median were closest to that of measured values among the three prediction methods for the 54 validation samples (Table 2). The ME and RMSE of 54 validation samples that collected in 2004 by OKLG ? TIN were much less than that by OKRK and OKNS, which demonstrated that OKLG ? TIN was better than OKRK, OKNS in

1101

the interpolation of severely skewed data with several high peak values. In general, kriging could provide an unchanged mean, and all three of the back-transformation methods used in this study yield a prediction of a median rather than the mean for each kriged location. So, all the three medians were close to that of the measured value. In most cases, the ME and RMSE of 20 randomly selected validation samples by OKLG ? TIN were also less than that by OKRK and OKNS. However, the ME and RMSE of 20 randomly selected validation samples by OKLG ? TIN were closed to that by OKRK and OKNS when high peak values were selected for validation. It indicated that the local variability in the neighborhood of high peak value will be concealed when it was excluded for prediction data and it could not be predict based on holistic spatial variability. There was a distinct difference in the predictions of high peak values by different interpolation methods (Fig. 4). For those high peak values, the predicted by OKLG ? TIN were very close to their measured values, while that by OKRK and OKNS were far less than their measured ones (Fig. 5). This demonstrated that OKLG ? TIN could predict the high peak values successfully, but not by OKRK and OKNS. All evaluation indicators of OKLG ? TIN were better than that of OKRK and OKNS except the maximum and standard deviation when the four high peak values were excluded from the samples for validation (Table 2). It demonstrated that the integrated method improved the prediction precision of other data again. The major reason was the prediction based on OKLG ? TIN was not influenced by high peak values as we have mentioned in front text. To the data set B that could reflect the most importance local variability of hot spots of soil contaminate i.e. the neighborhood of high peak values approximately. However, the sampling density in the neighborhood of high peak value was always too sparse to know the detail characters of local spatial variability IDW is based on the assumption that the interpolating surface should be influenced most by the nearby points and less by the more distant points. This assumption was always consisting with the fact of pollutions distribution, and IDW was popular in pollutions spatial prediction of single pollution source. Whereas, there were several high peak values that come from different pollution source in contaminated site and their affect ranges had large difference. IDW with high power often increase the error in the neighborhood of high peak values for it make the predicted value decrease sharply, and IDW with low power often increase the error in the locations of non-peak value for it had wide influence range. So, it is difficult to choose a fitted power for IDW interpolation. Therefore, it was difficult to choose a uniform appropriate power parameter. TIN is also one of the most commonly used techniques for interpolation of scatter points, and the prediction was only influenced by the values of three nearest known

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points. Although the effect of TIN interpolation very much depends on the density of observations TIN maybe not the best method of conventional interpolation methods for data set B, and further research should attempt to look for a more suitable interpolation method for data set B. In this study, TIN was used for it seldom even to does not smooth the spatial variation of Cu estimates for the prediction was only influenced by the values of three nearest known points, whereas the estimates by other interpolation method are weighed linear combinations of several sample values, and hence overestimation of small, and underestimation of large, values occur. In this study, to majority samples for validation, the predicted values based on three methods were close to the measured value and had no significant difference for they were not in the neighborhood of high peak values and the effect of high peak values on their prediction were limited (Fig. 5). It demonstrates that the prediction precisions of OKLG, OKRK and OKNS for the same data had no large difference if the data satisfy their requirements, and the study of Wu et al. (2006) also demonstrated it. So, it was unnecessary to compare the spatial prediction by OKLG ? TIN with these by integration of OKRK and TIN and by integration of OKNS and TIN, respectively, for the data set A satisfies lognormal distribution. OKLG ? TIN was better than the OKRK and OKNS in spatial interpolation of severely skewed data with several high peak values, in large part because the data partition improve the suitability for kriging and alleviate the smooth effect of high peak values in kriging, and TIN interpolation could describe the local variability around high peak values. Transformation and back-transformation may have other effects that are hard to interpret or may add uncertainty (Armstrong and Boufassa 1988; Journel and Deutsch 1997; Deutsch and Journel 1998; Roth 1998; Goovaerts 1999; Jordan et al. 2007). The integrated method of kriging and TIN could extend the scope of application of kriging interpolation, and it was suitable for spatial interpolation of severely skewed data with several high peak values. High peak values were often presumed to come from a different distribution than the majority of the survey data in environmental study, and they often reflect local variability. The common survey data were often having a global variability. It indicated that it is necessary to deal with the high peak values alone when we spatially interpolate the environmental survey data with several high peak values. Spatial interpolation methods can be classified in different ways and are generally defined based on geometric or geostatistical properties. Spatial interpolation methods can generally be classified as local or global, exact or approximate, and deterministic or stochastic methods. Local interpolation methods work on a small portion of the data points while global methods work across the whole

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data set (Ali 2004). To this criterion, OKLG is one of global interpolation methods, and TIN is one of local interpolation methods for TIN only use three values and kriging usually use much more samples in the study. Therefore, the integrating kriging and TIN has the potentials to describe both holistic and local variability of environmental survey data with several high peak values. OKLG and TIN maybe not the optimal method for data set A and data set B interpolation, respectively. However, the method of data split and integrating geostatistics and conventional statistic method had potential to improve the precision of severely skewed data with several high peak values.

Summary In case of severely skewed data with several high peak values, rank order and normal score transformations could reduce skewness, but both OKRK and OKNS could not make satisfied interpolations. The predictions by integrating OKLG and TIN had a smaller ME and RMSE than that of OKRK and OKNS. After the high peak values were substituted by the median of the remaining data set, the smooth effect could be alleviated sharply by OKLG and TIN interpolation, which provided more accurate predictions for the areas around the high peak values. This integrated method has the potentials to apply in other cases of severely skewed data with several high peak values. Acknowledgments This research was funded in part by the Science and technology support program Grant (2007BAC16B06) and the National Basic Research Priorities Program (973 Program) Grant (2002CB410810). We appreciate our staffs for soil sample collection and analysis. We also extend our appreciation to the journal reviewers and Editor-in-Chief, Dr. LaMoreaux for their valuable suggestions and constructive criticism.

References Ali TA (2004) On the selection of an interpolation method for creating a terrain model (TM) from LIDAR data. In: Proceedings of the American Congress on Surveying and Mapping (ACSM) Conference 2004, Nashville, TN, USA Andrews DF (1974) A robust method for multiple linear regression. Technometrics 16:523–531 Armstrong M, Boufassa A (1988) Comparing the robustness of ordinary kriging and lognormal kriging-outlier resistance. Math Geol 20:447–457 Arrouays D, Mench M, Amans V, Gomez A (1996) Short-range variability of fallout Pb in a contaminated soil. Can J Soil Sci 76:73–81 Atkinson AC (1994) Fast very robust methods for the detection of multiple outliers. J Am Stat Assoc 89:1329–1339 Atteia O, Dubois JP, Webster R (1994) Geostatistical analysis of soil contamination in the Swiss Jura. Environ Pollut 86:315–327 Baraba´s N, Goovaerts P, Adriaens P (2001) Geostatistical assessment and validation of uncertainty for three-dimensional dioxin data

Environ Earth Sci (2011) 63:1093–1103 from sediments in an estuarine river. Environ Sci Technol 35:3294–3301 Bardgett RD, Speir TW, Ross DJ, Yeates GW, Kettles HA (1994) Impact of pasture contamination by copper, chromium, and arsenic timber preservative on soil microbial properties and nematodes. Biol Fertil Soils 18:71–79 Cambardella CA, Moorman TB, Novak JM, Parkin TB, Karlen DL, Turco RF, Konopka AE (1994) Field-scale variability of soil properties in central Iowa soils. Soil Sci Soc Am J 58:1501–1511 Cao ZH, Hu ZY (2000) Copper contamination in paddy soils irrigated with wastewater. Chemosphere 41:3–6 Carlon C, Critto A, Marcomini A, Nathanail P (2001) Risk based characterisation of contaminated industrial site using multivariate and geostatistical tools. Environ Pollut 111:417–427 Cattle JA, McBratney AB, Minasny B (2002) Kriging method evaluation for assessing the spatial distribution of urban soil lead contamination. J Environ Qual 31:1576–1588 Das SK, Basudhar PK (2006) Comparison study of parameter estimation techniques for rock failure criterion models. Can Geotech J 43(7):764–771 Deutsch CV, Journel AG (1998) GSLIB, geostatistical software library and user’s guide. Oxford University Press, New York Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York Goovaerts P (1999) Geostatistics in soil science: state-of-the-art and perspectives. Geoderma 89:1–45 Hendficks Franssen HJWM, van Eijnsbergen AC, Stein A (1997) Use of spatial prediction techniques and fuzzy classification for mapping soil pollutants. Geoderma 77:243–262 Huang M, Shu YR, Huang DY, Wu JS, Huang QY (2006) An on-thespot sampling and survey method for soil nutrient cycling study (In Chinese). Chin J Appl Ecol 17(2):205–209 Isaaks EH, Srivastava RM (1989) Applied geostatistics. Oxford University Press, New York Jordan C, Zhang CS, Higgins A (2007) Using GIS and statistics to study influences of geology on probability features of surface soil geochemistry in Northern Ireland. J Geochem Explor 93:135–152 Journel AG (1980) The lognormal approach to predicting local distributions of selective mining unit grades. Math Geol 12:285–303 Journel AG, Deutsch CV (1997) Rank order geostatistics: a proposal for a unique coding and common processing of diverse data. In: Baafi EY, Schofield NA (eds) Geostatistics Wollongong ‘96. Kluwer, Dordrecht Juang KW, Lee DY (1998) A comparison of three kriging methods using auxiliary variables in heavy-metal contaminated soils. J Environ Qual 27:355–363 Juang KW, Lee DY, Chen ZS (1999) Geostatistical cross-validation for additional sampling assessment in heavy-metal contaminated soils. J Chin Inst Environ Eng 9:89–96 Juang KW, Lee DY, Ellsworth TR (2001) Using rank order geostatistics for spatial interpolation of highly skewed data in a heavy-metal contaminated site. J Environ Qual 30:894–903 Lee DY, Juang KW (2003) Use geostatistics to delimit the boundary of pollution in a contaminated site (In Chinese). Taiwan’s Soil Groundwater Environ Protect Assoc Newslett 7:2–13 Li GG (2005) The status and development needs of soil environmental monitoring in China (In Chinese). Environ Monitor Technol 17(1):8–10 Liu FC, Shi XZ, Yu DS, Pan XZ (2004) Mapping soil properties of the typical area of Taihu Lake Watershed by geostatistics and geographic information systems––a case study of total nitrogen in topsoil (In Chinese). Acta Pedol Sin 41(1):20–27 Loland JO, Singh BR (2004) Copper contamination of soil and vegetation in coffee orchards after long-term use of Cu fungicides. Nutr Cycl Agroecosyst 69:203–211

1103 Loska K, Wiechula D, Pelczar J (2005) Application of enrichment factor to assessment of zinc enrichment/depletion in farming soils. Commun Soil Sci Plant Anal 36:1117–1128 Lu Y, Gong ZT, Zhang GL, Burghardt W (2003) Concentrations and chemical speciation of Cu, Zn, Pb and Cr of urban soils in Nanjing, China. Geoderma 115:101–111 Luo Y, Jiang X, Wu L, Song J, Wu S, Lu R, Christie P (2003) Accumulation and chemical fractionation of Cu in a paddy soil irrigated with Cu-rich wastewater. Geoderma 115:113–120 Meuli R, Schulin R, Webster R (1998) Experience with the replication of regional survey of soil pollution. Environ Pollut 101:311–320 Moreno JL, Garcia C, Hernandez T, Ayuso M (1997) Application of composted sewage sludges contaminated with heavy metals to an agricultural soil. Soil Sci Plant Nutr 43:565–573 Mueller TG, Pusuluri NB, Mathias KK, Cornelius PL, Barnhisel RI (2004) Site-specific fertility management: a model for predicting map quality. Soil Sci Soc Am J 68(6):2031–2041 Park D, Cho H, Kim Y (2001) A TIN compression method using Delaunay triangulation. Int J Geogr Inform Sci 5(3):255–269 Rawlins BG, Lark RM, O’Donnell KE, Tye AM, Lister TR (2005) The assessment of point and diffuse metal pollution of soils from an urban geochemical survey of Sheffield, England. Soil Use Manag 21(4):353–362 Roth C (1998) Is lognormal kriging suitable for local estimation? Math Geol 30:999–1009 Saito H, Goovaerts P (2000) Geostatistical interpolation of positively skewed and censored data in a dioxin-contaminated site. Environ Sci Technol 34:4228–4235 Schwertman NC, de Silva R (2007) Identifying outliers with sequential fences. Comput Statist Data Anal 51:3800–3810 Schwertman NC, Owens MA, Adnan R (2004) A simple more general boxplot method for identifying outliers. Comput Statist Data Anal 47(1):165–174 State Environmental Protection Administration of China (1995) Chinese environmental quality standard for soils (GB 156181995). http://www.chinaep.net/hjbiaozhun/hjbz/hjbz017.htm Tan MZ, Xu FM, Chen J, Zhang XL, Chen JZ (2006) Spatial prediction of heavy metal pollution for soils in peri-urban Beijing, China based on fuzzy set theory. Pedosphere 16(5):545– 554 Tukey JW (1977) Exploratory data analysis. Addison-Wesley, Reading van Meirvenne M, Goovaerts P (2001) Evaluating the probability of exceeding a site-specific soil cadmium contamination threshold. Geoderma 102:75–100 von Steiger B, Webster R, Schulin R, Lehmann R (1996) Mapping heavy metals in polluted soil by disjunctive kriging. Environ Pollut 94:205–215 Walter C, McBratney AB, Viscarra Rossel RA, Markus JA (2005) Spatial point-process statistics: concepts and application to the analysis of lead contamination in urban soil. Environmetrics 16:339–355 Wu J, Norvell WA, Welch RM (2006) Kriging on highly skewed data for DTPA- extractable soil Zn with auxiliary information for pH and organic carbon. Geoderma 134:187–199 Zhang CS, McGrath D (2004) Geostatistical and GIS analyses on soil organic carbon concentrations in grassland of southeastern Ireland from two different periods. Geoderma 119(3–4):261–275 Zhang CS, Selinus O (1998) Statistics and GIS in environmental geochemistry––some problems and solutions. J Geochem Explor 64:339–354 Zhang CB, Li ZB, Yao CX, Yin XB, Wu LH, Song J, Teng Y, Luo YM (2006) Characteristics of spatial variability of soil heavy metal contents in contaminated sites and their implications for source identification (In Chinese). Soils 38(5):525–533

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