use of spatial prediction techniques and fuzzy classification for mapping soil pollutants

ELSEVIER Geoderma 77 (1997) 243-262

GEODER~

Use of spatial prediction techniques and fuzzy classification for mapping soil pollutants

H.J.W.M. Hendficks Franssen a,*, A.C. van Eijnsbergen b, A. Stein a

a Department of Soil Science and Geology, Agricultural University, P.O. Box 37, 6700AA, Wageningen, The Netherlands

b Department of Mathematics, Agricultural Universi~, Dreijenlaan 4, 6703 HA, Wageningen, The Netherlands

Received 14 November 1995; accepted 27 February 1997

Abstract

High concentration of heavy metals caused by soil pollution processes vary in a three-dimensional space. This study focuses on a strip of land which was sampled at three different depths. Fuzzy classification by fuzzy k-means was used to analyze the spatial variability for each separate layer. The concentrations of heavy metals showed abrupt changes in geographical space and the coefficients of variation were high, but in most of the area more gradual transitions in concentrations of heavy metals were found. To avoid undesirable fragmentation in the classification, i.e. a lot of small pieces of land, spatial variability is used by including additional predicted values of Cu, Pb and Zn representing spatial correlation in the horizontal and vertical plane. Membership values are interpolated for each class by ordinary kriging. For all situations the different maps are integrated into a final classification map with associated impurity map. The effect of observation density, class uncertainty and fuzziness exponent is expressed on the impurity map. A comparison is made between fuzzy classification and probability based mapping by indicator kriging.

Keywords: fuzzy classification; spatial variability; kriging; map impurity; soil pollution; heavy metals

I. Introduction

Since the IP7Os, soil pollution studies have received considerable attention. If critical levels are exceeded, soil pollution may affect human health and life of biological

* Corresponding author. Current address: Departamento de Ingenierla Hidrfiulica y Medio Ambiente, Universidad Polit6cnica de Valencia, C/Camino de Vera s /n , Apdo. Correos 22012, 46071, Valencia, Spain.

0016-7061/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S001 6 - 7 0 6 1 ( 9 7 ) 0 0 0 2 4 - 4

244 H.J. W.M. Hendricks Franssen et al. / Geoderma 77 (1997) 243-262

organisms. Pollutants moving through soil may reach ground- or surface-water after some time (depending on soil characteristics) and directly influence health of living organisms by water uptake. Accumulation of toxic materials in plants later results in movement of these materials in the food chain. Further, volatilization of pollutants from soil may harm human health, for example by accumulation in basements of houses (De Haan et al., 1984).

In general the contents of pollutants may show complex spatial patterns. High peak values are found and mostly coefficients of variation are above one. Therefore, it is difficult to detect the areas with pollution above a critical level, even if data are collected at a large number of observation points. Furthermore, decisions on whether to remediate certain areas of land are associated with large uncertainty. Geostatistical prediction methods can be used in order to give an unbiased prediction with minimum variance for the content of a given pollutant (Stein and Corsten, 1991). However, because of the normally high coefficient of variation stable estimation of the semi-variogram is difficult and kriging standard deviations could be high (Journel, 1983). Lognormal kriging and indicator kriging use nonlinear transformations of the data to handle nonnormal data and peak values (Deutsch and Joumel, 1992, pp. 71-72). But even with these interpolation methods the uncertainty in predicting the probability of exceeding intervention levels for soil pollution may not be satisfactory. Therefore interpolating using fuzzy classification (McBratney and De Gruijter, 1992; Odeh et al., 1992) has been examined in this study.

Fuzzy classification recognizes transitions, e.g., at the field scale, where observations may partially belong to different classes (Burrough, 1989). As such it differs from hard classification, where observations are members of a single class. Fuzzy classification can also deal more easily with transitions in the areas which seem to be less gradual than kriging. These boundaries may occur where pollution is concentrated in a single spot. It has been observed in the past that classification gives better predictions than ordinary kriging in the presence of sharp boundaries, in particular if it was important to include information on spatial correlation in the classification (Voltz and Webster, 1990). Individual sites are grouped that exhibit the same properties and are taxonomically close to each other. However, small areas of land with different properties compared to the surrounding land will appear in the final classification (Oliver and Webster, 1989). In such situations inclusion of additional values, which take into account the spatial correlation, may yield smooth fuzzy classification required for remediation practices where undesirable fragmentation is to be avoided.

In our study area, different heavy metals are remediated in a single process. This therefore requires a multivariate classification. Fuzzy classification has been used to classify soil in a strip of land (spatial extensions 250 × 2000 m) contaminated with heavy metals in an urban area in the Netherlands. Information on spatial correlation is used by including the predicted values for the pollutants based on neighbouring points or layers. Prediction was carded out based on spatial correlations in the horizontal and vertical plane. Specifically, the aims of this study are: (1) to use a multivariate fuzzy classification on the environment data; (2) to combine spatial variability with classification in order to avoid undesirable fragmentation in the spatial layout; (3) to make a comparison of fuzzy classification maps and indicator kriging maps in order to evaluate

H.J.W.M. Hendricks Franssen et al. / Geoderma 77 (1997) 243-262 245

differences between a fuzzy and a probability based approach, i.e. to what extent the spatial patterns in the fuzzy classification maps are similar to found spatial patterns in the different indicator kriging maps.

2. Methods and materials

The study area is rectangular, 250 × 2000 m, located in an urban site in the western part of the Netherlands. The area is polluted with lead (Pb), copper (Cu) and zinc (Zn). Pollution originates mainly from the so-called anthropogenic layers, which developed from the late middle ages to the beginning of this century whereby fertilized peat soils were mixed with waste products, mud and manure. In sections of the area additional pollution occurred during the 20th century. A cable factory added pollutants Pb, Cu and Zn, by spreading these metals during coating of electric wires. In other parts of the area accumulation of waste material of unknown origin was found. Contents of lab, Cu and Zn were determined using atomic adsorption spectrometry.

It is likely that mass transport in the vertical plane causes a raise of Pb, Cu and Zn contents at deeper layers after some time. In general it can be assumed that heavy metals are subjected to a net flow downwards. Therefore, metal concentrations may have a high correlation with the contents of these heavy metals higher in the profile. Because of spatial variability in soil physical properties such as hydraulic conductivity, water flow is in winding stream patterns. A value at a specific depth in the profile is influenced by a small area in the layer above the point. To take this into account, Layer 2 values at observation points were predicted from Layer 1 using block kriging. Similarly, Layer 3 values were predicted from both Layers 1 and 2. Classification of Layers 2 and 3 is carried out by adding values which are based on spatial correlation in the vertical plane for an area of l0 X 10 m above this point.

Soil spatial variability in the horizontal plane was also taken into account because Pb, Cu and Zn contents were predicted at the rneasurement locations by skipping the measurement itself and kriging a value at this location by using the surrounding measurement data. Additional measurement data have been used for this spatial prediction; for the spatial prediction also locations could be used where only one or two of the variables were known, while for the data set used in the fuzzy classification for all variables values have to be known. The spatial prediction has been carded out for all observation points and all variables, for each layer separately. As a result the data set of each layer consists of both measured and predicted contents of Pb, Zn and Cu at these locations by means of a spatial interpolation based on the spatial correlation in the horizontal plane. As a result, variables based on spatial correlation are added_ so that undesirable fragmentation in classification is avoided and results are more smoothed. At each prediction point 12 surrounding observation points within a radius of I00 m were used to krige the value at the point, using an estimated semi-variogram (Balog, 1993, Appendix 5).

2.1. Fuzzy classification

Fuzzy classification (Burrough et al., 1992; McBralney and De Grnijter, 1992; McBratney et al., 1992; Odeh et al., 1992; De Gruijter et al., 1997; Lagacherie et al.,

246 H.J.W.M. Hendricks Franssen et al. / Geoderma 77 (1997) 243-262

1997) is used in this study to classify soil pollution data. From an earlier study (Balog, 1993, Appendices 10-15) gradual changes were observed, hence it requires a geostatistical approach or a fuzzy statistical classification technique. On the other hand, the presence of a clear separated pollution violated an assumption of stationarity, hence hampering the use of kriging.

To briefly describe fuzzy classification, suppose that the data to classify, sampled at n locations with p variables, determined at each location, are summarized in a matrix Y of size n by p. A location may partially belong to different classes and the degree of participation of location i to class c is quantified by a membership value, mic, which takes a value between zero and one. It is assumed that the sum of membership values at a location (row sum) is equal to 1 and that empty classes do not exist. Classification can be carried out by searching matrix M of membership values that minimizes an objective function that characterizes the badness of the classification. In fuzzy classification with fuzzy exponent ~ (q~ > 1), which can be chosen by the user, the objective function is a generalization of the within-class sum of squares of the matrix Y, defined by:

J ( M ) = mine ~,i Y~cm~d2( Y~,Zc) (1) where Y, is the ith row of matrix Y and the minimizing Z,. is the so-called fuzzy centre of class c and where d is the distance between two p-dimensional points. For q~ = 1 minimization with respect to the matrix M results in a matrix of a hard classification. For q~ > 1 minimization results in the following formulae for membership values and class centres:

d ~ 2 / ( ~ - 1)

mic = k l < i < n , l < c < k (2) ~., d~. 2/(~- ~)

c = l t/

~p Em. Zc = i=1 n l < _ _ c ~ k ( 3 )

Em . i=1

The class centre is a weighted average of the observed values at the points which participate in the class for all variables each. The weights are determined by the membership values raised to the power q~. If q~ equalizes 1 the classification is hard. In case of increasing q~, the relative participation of high membership values increases, yielding a minimum for J(M) for a more fuzzy distribution of membership values. If q~ is very large, observations tend to have an equal weight over all classes, yielding a high degree of fuzziness. Further, the value of J(M) is higher if the distance between observation point and class centre is higher. If the ~value increases higher membership values will be penalized more severely. For very high ~values this results in membership values as big as 1/k for all point in all classes. By means of a resulting Picard iteration the minimum value of J(M) can be obtained. The total computational technique includes seven steps (e.g. McBratney and De Gruijter, 1992).

In this study, a fuzzy classification has been carried out for each layer for k = 4 and for two values of the fuzzy exponent (q~ = 1.30 and 1.45). Four classes were sufficient

H.J. 14I.tl4. Hendricks Franssen et al. / C, eoderma 77 (1997) 243-262 247

(a) Sampling data of Cu, Pb, Zn and d.m.,

taken at three different depths

At the measurement locations Pb, Cu and Zn are also predicted, based on spatial correlation in the horizontal plane (measurer value

is skipped alad surrol111ding m~Isu-v~-t data, including extra data, are used for spatial

prediction)

At the measur~-t locations in the Layers 2 and 3 additional predicted values of Pb, Cu and Zn

are added, based on spat/al correlation in the vertical plane (Pb, Cu and Zn contents frt-, the

more intensively sampled Layer(s} above are used}.

For each layer a fuzzy classification is carried out.For Layer 1 7 variables, Layer 2 10 variables

(three add/tional variables based 0mvertical spatial correlation) , Layer 3 13 variables (six od~Jtional variables based on vertical spatial

correlation) are used.

Or~ ---y kriging has been used to intarpolate

the o b t a / n e d membership vl l lues for each class

and for each I~yer.

For each grid cell the dominant class is determined and

the ~ity. For each Layer final classification maps and

impurity maps are obtained.

Fig. 1. (a) Flow diagram of the main steps in the classification of the soil pollution da!~ (b--j) Experimental and fitted semi-variogram models for constructing artificial variables. (b) Layer-I Cu, (c) Layer-2 Cu, (d) Layer-3 Cu, (e) Layer-I Pb, (f) Layer-2 Pb, (g) Layer-3 Pb, (h) Layer-1 Zn, (i) Layer-2 Zn, (j) Layer-3 Zn.

to make a clear division without much loss of information. The values for the fuzzy exponent are in agreement with those in other studies (McBramey and De Gruijter, 1992; McBramey et al., 1992). Membership values for all observation points in the


(13) ,.oot~.os

3,00~÷05

2,00E+05

1.001:+05

0.01~+00

N 4,00E+06

3,00E+06 m

..L ~ 2.~E+~

1.00E+06

0 40 80 120 160 200

distance (metem)

0.00E*O0 0 40 80 120 160

dlsumce(met~)

Fig. 1 (continued).

different situations were obtained using the program MacFuzzy (Ward et al., 1992). The GSLIB-package was used for interpolating the obtained membership values (Deutsch and Joumel, 1992, pp. 91-93).

The calculated membership values are interpolated by ordinary kriging. The predicted membership values have also been used for making a final, hard, classification. In this case a point is allocated to the class with the highest membership value. The estimated final class CF(X 0) to which a certain grid point x o belongs is given by the formula:

cF(Xo) = argmaxc(m(x0) c) (4)

H.J. W.M. Hendricks Franssen et al. / Geoderma 77 (1997) 243-262

(el) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249

|

(e)

O.OaE~O

0

4.00E+OS

40 8O 120

~ (nmm) 1tl0 200

3,00E, t .~

i i 2.00E~

[

0,00S*00 0 40 80 120

~-~..,, (nmm)

Fig. 1 (continued).

120 200

where m ( x o ) ¢ denotes the kriged value of the membership to class c for grid point x o.

The uncertainty associated with this classification is defined as the map impurity:

M.I . (xo) = 1 - max(m(xo )c ) (5)

where m ( X o ) c denotes the kriged value of the membership to class c, c = 1 . . . . . 4 respectively for grid point x 0 (Bierkens and Burrough, 1993a, b). The index M.I.(xo) is 0 if a grid point x 0 is classified 'hard' in one of the four classes and 0.75 if membership values are the same for the four classes. The higher the fuzziness coefficient, the higher the average M.I.-value for the map. With fuzzy classification it is possible to make a


( f ) a.oaE~a5 , ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a

i

0.00E~O 0

8,00E~'.05 •

50 100 150 200 250

dls~,.,,, (nHdem)

300 3,~ 400

i 4.0E..~, E

2 ,~ , t , l~ ,

0 40 80 120 160 ~ 0

d~tm~e (mote~)

Fig. 1 (cont inued) .

division into classes while still having an index of uncertainty. Fig. la summarizes the whole followed approach. Other possible membership functions for fuzzy sets, a-priori membership functions in the sense of Burrough (1989), were not considered.

Hard classification can be considered as a special case of fuzzy classification. A hard classification of n locations into k classes can be characterized by an incidence matrix M = mic of size n by k; the value of mic is 1 if location i belongs to class c and 0 if location i does not belong to class c. The fact that each location is classified requires


(11) 2..++o5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

6 1,sates s

g

Z

(J)

o,oos,.oo

2.,,,~E,a)s

40 so 120 • ~ ~ )

160 200

2.00E*0~

s | el

i 1.00E~ w

U0E~4

0,OtE~ 4O

~ t m m , )

Fig. 1 (continued).

W

that the sum of mic for each site i (row sum) is equal to 1. Empty classes do not exist, hence each class has at least one participation.

2.2. Indicator kriging

Exceedance probabilities of soil pollution levels for copper, zinc and lead contents in Layer 1, Layer 2 and Layer 3 have been calculated with indicator kriging (Journel,


(j)

S

| a

0 40 80 190 160 200

di~ma* ( m . ~ )

Fig. 1 (continued).

1983). These maps were used for a comparison with maps obtained by the fuzzy k-means method.

3. Results

3.1. The data set

The data set, derived from several soil pollution investigations, consists of two coordinates (NW-SE direction: x-coordinate; NE-SW direction: y-coordinate) of the observation points and the contents (mg/kg) of Cu, Pb and Zn of samples at the observation points in the three layers (0-0.5 m, 0.5-1.0 m and 1.0-1.5 m). The Cu, Pb and Zn contents are all chemically determined by atomic adsorption spectrometry. Although the variables show a lognormal distribution the data have not been trans- formed lognormally. The long tail of the lognormal distribution will force that a separate, smaller class of (very) high values for the heavy metal contents will be formed. Furthermore, class centres can be expressed in terms of arithmetic averages. Since fuzzy classification has not a clear link with probability theory it is not obvious to perform the lognormal transform. Variables in each layer are denoted by their chemical symbol, followed by the layer number. Dry matter (d.m.) contents are included as well. The d.m. content is also of interest for soil remediation practice; multiplication of the pollutant content per kg dry matter with the dry matter content of the soil gives the total amount of pollutant.

In case of observation points with exactly the same coordinates (descended from different soil pollution investigations) the coordinates which belong to the observation


Table 1 Summary statistics for the measured variables in the fuzzy analysis

253

Layer Variable N Mean SD a CV b Minc Max (rag kg- i ) (rag kg- J ) (rag k g - J ) (mg k g - i )

1 Cu 371 182 345 1.9 8.33 3895 Pb 371 498 553 1.11 8.33 4700 Zn 371 330 413 1.25 8.33 3800 d.m. 371 64.6 11.4 0.176 33 94.9

2 Cu 225 342 1973 5.77 5 28000 Pu 225 518 725 1.4 8.33 4800 Zn 225 381 ! 137 2.98 8.33 16000 d.m. 225 42.8 18.9 0.44 4.8 93.4

3 Cu 65 266 1035 3.89 5 8300 Pb 65 515 602 1.17 8.33 2600 Zn 65 416 875 2.1 8.33 5100 d.m. 65 34.3 18.2 0.53 10 85.6

a SD = standard deviation. b CV = coefficient of variation. c 8.33 = value substituted for detection limit (0.667 times detection limit); see text.

from the latest survey were slightly changed (0.5 m) in order to avoid observations with the same coordinates. This is realistic, because in practice it is nearly impossible to return to exactly the same location and (small) errors are naturally occurring when determining coordinates. In some cases measured contents of heavy metals were below the detection limit. The detection limit (12.5 mg/kg for all heavy metals) was replaced by a simple substitution method; all detection limit values were replaced by 0.67 times the original detection limit value. This is an appropriate value in case of a lognormal distribution.

To carry out fuzzy classification a full data set per layer is required, yielding 371 observation points for Layer 1,225 for Layer 2, and 65 for Layer 3. Summary statistics for these variables are given in Table 1. High concentrations are observed: the average Pb contents (498-518 mg/kg) are nearly as high as the soil pollution intervention value (530 mg/kg). Coefficients of variation exceed 1.0 in all cases, whereas for Layer-2 Cu it is almost equal to 6. Probability plots (not shown) also indicated skewed distributions with some large outlaying values. These show the highly variable nature of the underlying phenomena.

Additional predicted values of Cu, Pb and Zn in the observation points obtained by prediction based on neighbouring points in the horizontal planes and neighbouring points in the vertical plane have been created. In Table 2 and Fig. lb - j the semi-variogram models used for predicting the additional values of Cu, Pb, Zn are shown. The predicted values for the variables based on spatial correlation in the horizontal plane had the same averages as the measured values for the variables, but standard deviations were reduced noticeably because of 'smoothing' properties of kriging. Predictions were relatively poor for Layer 3 (for two of the variables the linear correlation coefficient between the measured values and the predicted values is below 0.10) because of the

254 H.J.W.M. Hendricks Franssen et al , / Geoderma 77 (1997) 243-262

Table 2

Fitted semi-variogram models with the corresponding semi-variogram parameters for the nine variables

according to Balog (1993, Appendix 5)

Layer Variable Model Nugget Sill 1 / 3 X Range Slope (mg 2 kg - 2 ) (mg 2 kg - 2 ) (m)

(x 10 5 ) (x 10 5)

1 Cu Linear 1,16 - -

Pb Exponential 1,40 3.07 52.

Zn Exponential 7.50 8.62 26

2 Cu Exponential 0 55.2 15.7

Pb Exponential 3.61 5.62 146

Zn Exponential 0 21.4 20.4

3 Cu Exponential 0 49.6 409

Pb Linear 1.72

Zn Linear 8.48 - -

8560

81300

90800

small number of measurements in this layer (see Table 3). The correlation coefficients between measured and predicted values are relatively high for predictions based on spatial correlation in the vertical plane. However, the correlations between the spatial predictions in the horizontal plane and the original measurements are lower. Especially for Layer 3 predicted values and measured values are only slightly correlated. However, the observed linear correlations support inclusion of these variables in the fuzzy classification.

Table 3 Linear correlation coefficients between measured contents and predicted contents

Layer Variable H a V1 b V2 c

1 Cu 0.20 - -

Pb 0.48 - -

Zn 0.47 - -

2 Cu 0.01 0.47 -

Pb 0.42 0.56 -

Zn 0.16 0.38 -

3 Cu 0.06 0.41 0.64

Pb 0.3 l 0.46 0.46 Zn - 0.04 0.71 0.43

a Horizontal: correlation in the single horizontal plane between measured contents and predicted contents by kriging. b Vertical 1: correlation between measured contents and the predicted contents by means of information of the soil layer above.

Vertical 2: correlation between measured contents and the predicted contents by means of information of

two soil layers above.


3.2. Fuzzy classification and interpolation of membership values

Fuzzy classification has been carried out using the original and the predicted values of the variables. Because of the relatively high correlations between these, the Maha- lanobis distance has been used. Grouping into four classes was the most suitable; it was thought that a division into five or more classes did not make the resulting classes mappable and a division into three classes did not separate the class centres enough. A fuzzy classification with ~ = 1.30 was chosen as the main situation. Using a fuzzy exponent lower than 1.30 yielded a relatively harder classification; a fuzzy exponent of 1.45, on the contrary, yielded classes that are very fuzzy (especially for Layer 3). It seemed that the maps obtained with ~0 = 1.30 were of a desired fuzziness. Furthermore, according to Ward et al. (1992), q~ has to be equal to 1.30 or smaller if the Mahalanobis distance is used. Results of calculations with ~o = 1.30 and 1.45 are presented.

For each layer a clear sequence in the four calculated class centres was found; one class (C l) with low values for all metals, and one class (C 4) with high values for all metals. The other two classes (C 2 and C 3) were intermediate (see Table 4 for the class centres for Layer 1). Due to the moderate strong linear correlations between most of the variables a clear distinction is found between a class with low values for all variables (C]) and a class with high values for all variables (C4). The membership values for C], C 2, C 3 and C4 were interpolated, yielding four maps for each layer. Interpolations were carded out with ordinary kriging, although McBratney et al. (1992) recommend log-ratio transformation prior to kriging followed by back transformation. First semi-variograms were estimated. The experimental semi-variograms have been fitted to a linear, spherical, exponential and Gaussian semi-variogram model and the best fitting model has been used to interpolate the membership values (see Table 5) (McBratney and Webster, 1986). In all cases except one, the best fitting model was a spherical or exponential one. The ranges are longer for C~ (172-578 m for the three layers) than for C 4 (25-126 m for the three layers), whereas the nugget effect is small for all the classes; in nearly all cases the nugget effect is less than 0.02. The difference in range values between e.g. C 1 and C 4 can be understood taking into account that membership values for C a are (very) low over most of the strip of land, and only in two relatively small spots higher membership values are found. The range value of the membership values for C 4 is therefore relatively small.

The maps in Fig. 2a indicate that for Layer 1 high membership values for C 3 and C 4 occur in two spots. The membership values for C l and C 2 are above zero everywhere.

Table 4 The class centres for Layer 1 as obtained with fuzzy classification (~o= 1.30; a subscript h indicates horizontally predicted variables; see text)

Class Cu Pb Zn d.m. Cuh Pb h Zn h (mg k g - 1 ) (mg k g - J ) (mg kg - I ) (rag kg i ) (mg k g - I ) (rag kg - i )

C r 106 250 165 76 99 266 147 C 2 128 382 204 56 113 383 209 C 3 202 576 459 63 198 596 530 C a 318 933 607 68 340 1052 628


(a) x-coordbmte (m)

2000 -

C l C2 C3 C4 N

/ t

1500-

1 0 0 0 -

soo.

o'

Membersh ip va lues

- 1.00

- 0.90

- 0 .80

- 0.70

- 0.60

"0.50

0.40

0.30

- 0.Z0

"0 .10

', 0.00 i I I

o 2,0 0' 250 0' • 2so 0' d0

(b) x-coordinate (m)

2000

Cu Pb Zn / N

1500

1000

500

|

q a

m m | 5=0 it m 0 250 0 2 0 250

y-coordinate (m)

Probabi l i ty

1 .oo

0.90

0.80

' 0.70

" 0.60

• 0 . 5 0

• 0.40

- 0.30

0.20

0 . I 0

• 0.00

Fig. 2. (a) Maps showing the interpolated membership values for Layer 1 for C 1, C 2, C 3 and C 4. (b) Probabilities of exceeding intervention values of soil pollution obtained by indicator kriging for Layer l for Cu, Pb and Zn.

H.J.W.M. Hendricks Franssen et al. / Geoderma 77 (1997) 243-262

Table 5 The semi-variogram parameters of the fitted models for each class

257

Layer Class Model Nugget (rag 2 kg -2) Sill-nugget (rag 2 kg -2) Range(m)

l C I Spherical 3.40× 10 -2 2.00)< 10-l 578 C 2 Spherical 1.81 × 10 -2 1.51 × 10- l 125 C a Spherical 8.7,1× 10 -3 1.40× 10-J 1000 C 4 Spherical 1.67 × 10- 2 8.18 × 10- 2 126

2 C I Exponential 2.40× 10 -2 1.43 X 10- t 360 C 2 Exponential 1.59× 10 -2 1.48× 10 -2 161 C 3 Exponential 1.67× 10 -2 8.51 × 10 -2 36 C a Exponential 0 7.28 × 10- 2 25

3 C t Exponential 3.58× 10-3 1.24× 10-J 172 C 2 Gaussian 2.03 × 10 - 4 1.3 × 10- 2 15 C 3 Exponential 0 6.14× 10 -2 36 C a Exponential 9.41 × 10- 3 3.73 × 10- 2 69

The maps for Layers 2 and 3 show similar patterns, but less obvious compared with Layer 1. The patterns of C 1 and C4 are inverse of each other. In all layers the same two spots of soil pollution are detected by means of high membership values for C 3 and C 4. Furthermore, for most class centres the measured and predicted contents of a certain variable were very close. Therefore it is likely that the artificial variables based on spatial correlation in the lateral and vertical plane avoided an undesirable fragmentation. The kriging variances are quite high, again indicating strong spatial variation. Further- more, three maps were produced which indicate the predicted exceedance probabilities of the intervention value of soil pollution obtained by indicator kriging (Journel, 1983) for Cu, Pb and Zn in Layer 1. A global comparison can be made. The indicator kriging maps indicate the occurrence of two pollution spots for Cu, Pb and Zn (see Fig. 2b). The same spots are found in the fuzzy classification as high membership values (> 0.25) for C 4. The indicator kriging maps also indicate that the polluted area in the northwestern part of the area is more spot-like and that the one in the southeastern part of the area is more extended. This pattern can also be distinguished in the fuzzy classification maps; the spot in the southeastern part is surrounded by an area with high membership values for C 3. The area with low exceedance probabilities ( < 0.10) of the intervention values for Pb,Cu and Zn has high membership values ( > 0.5) for C 1. It is clear that the maps resemble quite well and that with fuzzy classification an adequate statistical description of a multivariate soil pollution can be obtained.

3,3. Final classification and map impurity

The different maps are integrated into one final map. The heavily polluted spots are clearly visible because the dominant class is C 4 or C 3 (Fig. 3a). The map impurity is highest in the transition zones between two dominant classes. Also within the classes the map impurity is above zero which reflects the internal spatial variability (Fig. 3b).The


( a )

x-coo, C~,r,,~e (m

2000 -

t I soo.

1000-

500.

0 -

Laver I Layer 2 Layer 3

I • 01 I o ~o o' ~'o ~o

/ Class

F-1

N

(b) x-coordinate (m)

2000.,

t 1500,

1000-

SO0.

Layer I Layer 2 Layer 3

I " I r i 0 250 0 250 o' 2~'0

~ y--e~'d~te (m)

N

/ Map impur i ty

- 1 . o o

- 0.90

- 0.80

" 0.70

- 0.60

"0.50

- 0.40

- 0.30

- 0 . Z 0

-0 .10

- 0.00

Fig. 3. (a) Final classification of Layer l, Layer 2 and Layer 3. (b) Map impurities for Layer 1, Layer 2 and Layer 3.


Table 6

Fraction of the grid points classified into a class for different layers and values for the fuzzy exponent q~

Laye r and fuzzy exponent Class Number of gfidpoints Percentage in class

Layer 1, ~0 = 1.30 C I 1970 43.9

C 2 1371 30.6

C 3 812 18.1

C 4 332 7.4

Layer 2, q~ = 1.30 C 1 1801 44.4

C 2 1992 40.2

C 3 510 11.4

C 4 182 4.1

Layer 3, ~p = 1.30 C t 1931 43.1

C 2 1588 35.4

C 3 616 13.7

C 4 350 7.8

Layer 1, q~ = 1.45 C 1 1818 40.5

C 2 1392 31.0

C 3 941 21.0

C a 334 7.4

Layer 2 q~ = 1.45 C I 1588 35.4

C 2 2•96 49.0

C 3 473 10.5

C 4 228 5.1

same classifications are also made for a fuzzy exponent of 1.45. Nearly the same final classification was found, but map impurities were clearly higher because of the higher fuzziness exponent (see Tables 6 and 7).

The percentages of the different classes in the area can be displayed. This has been done for five situations. For all layers and both the two fuzzy exponents, 4 to 8% of the area was classified as heavily polluted (C 4) (see Table 6). Table 7 displays the average map impurities. The average map impurity denotes that the uncertainty (impurity) is higher in areas with a low number of observations. The role of the fuzzy exponent is also expressed; for both Layers 1 and 2 the map impurity was higher for q~ = 1.45 than for q~ = 1.30 (0.366 versus 0.323 and 0.466 versus 0.421, respectively).

The concept of a final classification map with associated map impurity may be

Table 7

Calculated average map impurities for different layers and fuzzy exponent ~p

Layer and fuzzy exponent map Impurity

Layer 1, ~0 = 1.30 0 .323

Layer 2, ~p = 1.30 0 .392

Layer 3, ~p = 1.30 0 .466

Layer 1, ~0 = 1.45 0 .366

Laye r 2, ~ = 1.45 0.421


attractive for a land user. If for example for a certain grid cell a membership value of 0.7 for C 2 and 0.3 for C 3 is found, this can be interpreted in terms of uncertainty (C 2 is the dominant class, but there is a probability of 0.3 that at a certain location the class is C 3) or spatial variability (C 2 is expected to be the dominant class for this land, but the total amount of inclusions of C 3 is expected to be 0.3 times the area surface).

4. Discussion

In this study we were confronted with two conflicting objectives. One objective was to use spatial information to avoid the occurrence of sharp differences in classification at nearby sites. The other objective was to focus on the original measurements to represent the population as adequately as possible. In the actual classification the contribution of spatial information is higher when the spatial correlation is stronger and vice versa. We chose the weight of the spatial information in the classification (in terms of the number of additional predicted values at a location) intuitively, without using any optimization procedure. Then the influence of the additional values on the classification is of a more random nature. Further research is necessary to better accommodate to these problems.

Inclusion of additional values of Cu, Pb, Zn obtained by prediction both in the vertical and in the horizontal plane were helpful to use additional spatial information in the classification. Correlation in the vertical plane improves classifications for the second and third layer. Correlations within the soil profile seem even higher than correlations between closely located spatial observations in the horizontal plane. In a fuzzy classification only those observation points can be used where all variables have been recorded, whereas points where only one, two or three variables have been measured are excluded. In the predicted values for the variables, however, this additional information is included because for the geostatistical prediction at the selected locations all observations are used.

The fuzzy exponent expresses the degree of fuzziness of the classification and is vital for the final classes. In this study the fuzzy exponent value was chosen by the user, using recommendations from earlier studies, whereas calculations have been carried out for other values as well. However, also mathematically heuristic methods exist for determining an appropriate fuzzy exponent that are more objective than just arbitrary selection of fuzzy exponents (Odeh et al., 1992). It is highly important to find objective criteria for determining the value of the fuzzy exponent.

The question must now be faced whether fuzzy classification or stochastic techniques are to be preferred for environmental studies. In a sense, these are complementary activities: a stochastic approach assumes the presence of a random field, which can be modelled using observations and from which inferences can be made. In a sense, the assumption is arbitrary, since such a field has no direct physical meaning. The willingness to make such an assumption has led to the wide-spread use of kriging-like procedures. Moreover, in many statistical testing and estimating procedures some assumption on the underlying distribution is made and used to the advantage. On the other hand, a fuzzy classification classifies the observations into homogeneous classes,

H.J. W.M. Hendricks Franssen et al. / Geoderma 77 (1997) 243-262 261

allowing the presence of imprecise boundaries. No assumptions about dislributions need to be made. For fuzzy classification, personal judgement is still required at several stages: to select the appropriate degree of fuzziness, to define the appropriate distance measure, to select the number of classes and to arrive at one single map from the individual class membership maps.

In this study, we compared a stochastic technique, indicator kriging showing the probabilities of exceeding a critical level, with the fuzzy k-mean classification. As is obvious from this study, the fuzzy classification is useful if a multivariate classification of spatial data needs to be performed. Both gradual and abrupt changes occur in geographical space, leading to an advantageous use of fuzzy classification. Fuzzy classification offers a powerful, though rather simple, instrument to determine the internal variability of spatial classes (the map impurity).

As compared to indicator kfiging the results of a multivariate classification, though, are more difficult to interpret. The output is not in the form of an exceedance probability of a critical level for a univariate property, but in the form of different degrees of participation to more complex defined classes. If the difficulty of interpretation is a problem, stochastic techniques should be preferred. Also, if location-specific estimations are required or if exceedance probabilities of univariate properties have to be estimated, fuzzy classification is of limited help.

On the other hand, critical levels for soil concentrations are always defined for univariate properties (single species). A multivariate definition of critical soil pollution levels would be preferable from a practical point of view: if an area of land contains many contaminants just below the threshold value it is probably more hazardous than if a single property is just above this threshold value. We could see the advantage of fuzzy k-means classification in this respect.

5. Conclusions

Fuzzy classification is useful to perform a multivariate classification of soil pollution data. A multivariate classification of soil pollution makes practical sense because in remediation practice heavy metals can be removed in one process. Comparison of interpolated maps with membership values for the different classes and maps with exceedance probabilities of critical levels, obtained by indicator kriging, show that fuzzy classification comprises information, but that at the same time characteristic patterns of e.g. the most contaminated sites in the area are maintained. The general gradual spatial variation is honoured because of the fuzzy character of the classification, at the same time fuzzy classification can also deal more easily with sharper boundaries in geographical space.

In this study, additional values of Cu, Pb and Zn obtained by prediction have been used with the original measurement values in the fuzzy classification. By doing so, undesirable fragmentation of the classified land in small units is avoided, which is in accordance with soil remediation practice. Undesirable fragmentation will be more avoided in case of stronger spatial correlation.


Map impurity is useful to model uncertainty associated with classified geographical data. Analysis of these kind of geographical data can be easily performed with Geographical Information Systems.

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