interpolation techniques for forecasting soil heavy …

Namysłowska-Wilczyńska B., Rusak K. (2001). Interpolation techniques for forecasting soil heavy-metal pollution foci in mining areas.

In: D. Kereković, E. Nowak (ed.). GIS Polonia 2001. Hrvatski Informatićki Zbor – GIS Forum, Croatia, 204-213.

INTERPOLATION TECHNIQUES FOR FORECASTING SOIL HEAVY-METAL POLLUTION FOCI IN MINING AREAS

Prof. Barbara NAMYSŁOWSKA-WILCZYŃSKA, PhD, DSc., Wrocław University of Technol-

ogy (WUT), Institute of Geotechnics and Hydrotechnics, Geology Division; Katarzyna RUSAK, M.Sc., doctoral student, Wroc ław University of Technology, Institute

of Geotechnics and Hydrotechnics, Geology Division;

ABSTRACT A survey of interpolation techniques for determining soil pollution state spatial characteristics is

made. The survey covers both estimation methods: ordinary point and block kriging and simple inter-

polation techniques: the linear kriging model, the nearest neighbour technique, the inverse distances

technique and moreover turning bands conditional simulation. Soil cadmium content determinations

made for the Bukowno-Sławków area by the Environmental Research and Control Centre in Katowice

were used for the computations. The presented techniques have proven to be useful tools for identify-

ing geochemical anomalies in heavy metal (cadmium) concentrations. INTRODUCTION Estimation of the variation of measurable parameters and indicators commonly occurs in environ-

mental protection problems. The state and degradation of a particular element of the environment can

be assessed and short- and long-term environmental forecasts can be made on the basis of numerically

expressed environmental parameters [3]. Increasingly widespread environmental monitoring supplies large amounts of data which must be ar-

chived in digital form (databases) and processed fast. Therefore increasingly faster and more efficient

and accurate tools are needed to process and interpret the data and to represent graphically the ob-

tained results. Geostatistics offers a possibility of obtaining full information about an investigated

element of the environment. One of the more interesting proposals in environmental research are the

ordinary (point and block) kriging estimation methods which make up an original procedure for esti-

mating the mean values of parameters (taking into account their autocorrelation) and turning bands

conditional simulation. Besides the rather advanced kriging and simulation methods based on variability represented by a

variogram, i.e. a function illustrating the variation in the values of a parameter depending on the dis-

tance between the points at which the parameter is measured [6], other simpler interpolation tech-

niques are presented in this paper. They require only a definition of the dimensions of samples search

subarea to estimate an elementary grid node. The cadmium pollution state of the surface layer of soils

in the area of Bukowno and Sławków (Małopolska Province) is studied. The variation images obtained using the above-mentioned simple and fast interpolation techniques are

analyzed and compared with the results yielded by the more advanced (but more laborious and time-

consuming) kriging and simulation methods to determine the areas and limits of the applicability of

the described techniques and their practical usefulness. CHARACTERIZATION OF STUDIED AREA Since the studied area of Bukowno-Sławków, situated in the E part of the Silesian Upland, is rich in

natural resources (it is often referred to as a zinc-lead basin) its geographic, economic, ethnographic,

cultural and social environment has been undergoing transformations for centuries. It is home to the

Bolesław Mine&Metallurgical Plant. Administratively the area belongs to Małopolska Province. The

area’s geographical position is specified by latitude coordinates 500 16′ N and longitude coordinates

190 27′ E. From the E side it merges into the Cracow Upland, from N into the Częstochowa Upland,

whereby the land is endowed with exceptional and varied topographic features. From the E-N side one

can see the ridge of the Cracow-Częstochowa Jura and from the S side the Silesian Upland extends. A

pine-covered plain 5 km2 in area (formerly desert -like and called the Starczynowska Desert or the

Beggar’s Sea) being part of the Błędowska Desert adds beauty to this land. Three mountains (called

the devil’s mountains): Diabla Góra (the highest – 382 m a.s.l.), Kozłowa Góra and Buczyna adjoin

the plane. The Świnia Góra mountain (ending the Ząbkowicki Ridge) towers at the end of Bukowno.



2

The region is situated in the basin of the Biała Przemsza flowing through Klucze, Okradzionów and

Sławków. In Okradzionów it is fed by the Biała River originating from the Ponikowski adit. The

Sztoła River, supplied with waters from the mines, flows through Bukowno and after Sławków it

flows into the Biał a Przemsza. In the S part of Bukowno there are numerous ponds in the worked out

sandpits (the sand was used as filling in coal mines) and an extensive forest covering the reclaimed

sandpit area [5].

The soils in the region are poor. In the area of Bukowno and the Bolesław rural district, covering 10

484 ha (44.5% forest and 29% farmland), there are soils belonging to the following soil quality

classes: 8.5% soils of class III, 53.0% soils of class IV and 38.3% soils of class V and VI. The agricul-

ture in the region is of small significance. Nature compensated for the poor soils with underground

natural resources which for centuries have been the basic source of income for the local population [5]. SCOPE OF STUDIES Data from the monitoring of the soils in the former Katowice Province, conducted in the years 1982-

1996 by the Environmental Research and Control Centre in Katowice [13], were used as the input for

geostatistical analyses of the soil cadmium content variation in the area of Bukowno - Sławków. The database prepared for the geostatistical analysis contained X and Y coordinate values and soil Cd

content determinations (in mg/kg dry mass). The sample size was 116 samples. Geostatistical software

package ISATIS (ver. 3.4.1), purchased from the firm Geovariances and Ecole des Mines de Paris in

Avon Cedex by the Institute of Geotechnics and Hydrotechnics of WUT, was used for the computa-

tions [2]. First the basic statistical parameters such as: the mean value, the minimum value, the maximal value, variance, standard deviation, skewness, kurtosis and the coefficient of variation were calculated and a Cd content distribution histogram was drawn. Then the isotropic, empirical Cd content variogram was computed. The soil Cd content was estimated using point and block ordinary kriging. Then computa-tions needed for turning bands conditional simulation were performed. The results of the estimation were presented on raster maps of estimated Cd content averages Z* and estimation standard deviations

δk and on different maps of simulated Cd content values Zs. The maps show the spatial distribution of

cadmium content (after performed estimation of average values Z*) within the area covered by the environmental monitoring as well as the results of Cd content extrapolation outside this area. Several raster Cd content maps produced using the so-called fast interpolation (which does not require

computing empirical variograms and modelling their courses) are shown for comparison. RESEARCH METHODS The main aim of the research was to determine, using different interpolation techniques, the spatial

variation of cadmium content in a 0-20 cm surface soil layer in the area impacted by the mining and

processing of zinc and lead ores (Bukowno and Sławków). This was accomplished by applying ordinary (point and block) kriging. In this procedure, sampling

point measurements are used to estimate a given investigated parameter at unsampled points, assuming

that there is isotropy and no trend in the behaviour of the regionalized variable [12]. The estimated

averages, calculated for elementary blocks (block kriging) or grid nodes (point kriging), are weighted

averages obtained on the basis of a small set of averages (control points). The previously determined parameters of the empirical Cd content variogram model, approximated by

the complex structure of a linear model and a spherical model with the nugget effect taken into ac-

count, were used in the kriging computations. In the case of ordinary point kriging, the analyzed area

was covered with a 250 m × 100 m grid and estimation computations were performed for the nodes of

this grid. In the case of block kriging, computations were performed for 500 m × 500 m elementary

blocks, assuming a 3 × 3 discretization coefficient for each block. In all, there were 5856 elementary

nodes and 693 blocks. Estimated averages Z* and kriging estimation standard deviations δk were cal-

culated for particular nodes and elementary blocks. Neither the spatial image of the raster map of estimated averages and estimation standard deviations

nor the ranges of maximum values Z* and δk changed as a result of the use of blocks of different di-

mensions (250 m × 250 m and 250 m × 500 m) in the block kriging.



In the next stage of the soil surface layer Cd contamination analysis, turning bands conditional simula-

tion was used [1, 8, 9, 10, 11]. The method makes it possible to simulate numerically n-dimensional



3

realizations of a stationary random function characterized by covariance C. In this way realizations

Zs(x) of random multidimensional Gaussian function Zs(x) can be obtained. Each realization Zs(x)

yielded by turning bands simulation is a sum of 15 independent one-dimensional realizations [12]. Simulations were run for cadmium content. The original data were transformed into Gaussian vari-

ables and the transformation function (Gaussian anamorphosis) was fitted [12]. The transformed data

were used for the computation of the isotropic empirical variogram which was approximated by a lin-

ear model and a spherical model with the nugget effect included. Then 20 conditional simulations of

Cd content were run. The obtained results were reconverted to the original form. As a result, statistical

simulation maps were obtained. Finally, raster maps of soil Cd content were computed using fast interpolation. The following interpo-lation techniques are applied in the paper:

Inverse distances and squared inverse distances, where estimation is a linear combination of

neighbouring data. The weight attached to each information is inversely proportional to the

distance or the squared distance from the data to the target. If the shortest distance is below the

assumed threshold, then the value corresponding to the sample is a simple copy of the nearest

point target [2]:

Z*=∑λα Zα

α Nearest neighbours – estimation consists in copying the value of the defined sample nearest to

the target. Thus it is a linear combination of neighbouring data [2]:

Z*=Zα-neighbors

Linear model of kriging- where: λ i(i=1,....n) – weights assigned to neighbouring sample data

n

Z*= ∑λ i Z(xi) i1

SPATIAL ANALYSIS OF SOIL CADMIUM POLLUTION STATE BY SELECTED INTERPOLATION TECHNIQUES

Ordinary kriging The ordinary kriging computations, aimed at obtaining the best estimate of the regionalized variable in

the unsampled places, were performed using the parameter values of the empirical cadmium variogram

approximated by the complex model structure (Fig.1.). Distance (Meter)

0. 1000. 2000. 3000. 4000.

150. 150.

: Cd

100.

100.

Variogram

405446 467

467 373

M1

452

459D1

263

50. 158

50.

0. 0.

0. 1000. 2000. 3000. 4000. Isatis Rusak/bukowslaw

- Variable #1 : Cd

Experimental Variogram(s) : 1 direction(s)

D1 - Az= 0.00, Ay= 0.00, Ax= 0.00

Angular tolerance = 90.00

Lag = 412.25032m, Count = 10 lags, Tolerance = 50.00% Model : 3 basic structure(s)

S1 - Spherical - Range = 1824.1393m, Sill = 53.97 S2 - Nugget effect, Sill = 23.84

S3 - Order-1 G.C. - Scale = 4690.035m, Sill = 17.84



Fig.1. Empirical variogram of cadmium Cd content with fitted model.



4

• Point kriging One large centre of high (22 -28 mg/kg) Cd content, consisting of three smaller contiguous foci and

one focus situated somewhat apart, is conspicuous on the raster map (Fig.2.). Also two smaller centres

with a Cd content of 12-14 mg/kg are visible. Each of the pollution centres is surrounded by a aureola

of decreasing Cd contents. The lowest cadmium contents (2-4 mg/kg) occur in the western and north-

eastern part of the studied area (Fig.2.).

X (Meter) 260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter) 874000. 874000.

873000. 873000.

872000. 872000.

Y 871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000. 867000.

260000. 265000. 270000. 275000. Fig.2. Raster map of estimated averages Z* of cadmium Cd content (the picture on the basis of point

kriging).

The lowest values of (kriging) estimation standard deviation δk (5-6 mg/kg) are observed in the

densely sampled places (Fig.3.). As the distance from the soil sampling points increases, the δk values

increase substantially to over 11 mg/kg (Fig.3.).

X (Meter) 260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

Y (Meter) 874000. 874000.

873000. 873000.

872000. 872000.

871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000.

265000. 270000.

867000.

260000. 275000.

Fig.3. Raster map of values of estimation standard deviation δk of cadmium Cd contents (point

kriging).

• Block kriging The picture of the spatial distribution of soil Cd pollution foci (Fig.4.) only slightly differs from the

one obtained by point kriging (Fig.2.). The estimated averages of Cd contents are within a range of

2.93-27.47 mg/kg which is similar to the 2.75-28.94 mg/kg range for point kriging.



5

X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter)

874000. 874000.

873000. 873000.

872000. 872000.

Y

871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000. 867000.

260000. 265000. 270000. 275000. Fig.4. Raster map of estimated averages Z* of cadmium Cd content (the picture on the basis of block

kriging). Much more pronounced differences appear when one compares the raster maps for estimation standard

deviations δk (calculated using both techniques) (Fig.3. and Fig.5.). Block kriging makes its possible

to substantially reduce the estimation error for δk from 5-6 mg/kg to 2-3 mg/kg at densely sampled

places and from over 11 to 9 mg/kg in sparsely sampled areas (Fig.5.). This is confirmed by the results of the geostatistical studies of parameters variation for the polimetallic copper ore deposits in the Foresudetic Monocline reported in monograph [7].

X (Meter) 260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter) 874000. 874000.

873000. 873000.

872000. 872000.

871000.

871000.

Y

870000. 870000.

869000. 869000.

868000. 868000.

867000. 867000.

260000. 265000. 270000. 275000.

Fig.5. Raster map of values of estimation standard deviation δk of cadmium Cd contents (block

kriging).

Conditional simulation (turning bands) Using turning bands simulation the following statistical simulation maps were obtained: a map aver-

aged of 20 simulations, a map of the smallest and largest realizations and a map of realizations stan-

dard deviation values. Because of the narrow confines of the paper only the map averaged for 20 simu-

lations (Fig.6.) and the map of realizations standard deviation (Fig.7.) are presented. The image on the

map shown in (Fig.6.) differs quite significantly from the one obtained by means of (point and block)

kriging. This is due to the method’s mathematical assumptions – the aim being to supply a set of ap-

propriate values which represent the same variations as the true values do. The soil Cd pollution su-

bareas determined by simulation are larger and their boundaries less distinct than on the kriging maps



which show smoother boundaries. The images obtained by conditional simulation are characterized by

a high degree of detail – several smaller foci of elevated Cd content beyond the main pollution centre



6

have been revealed. They are not visible on the kriging raster maps of estimated averages (Fig.2. and

Fig.4.). X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter

)

874000. 874000.

873000. 873000.

872000. 872000.

Y

871000. 871000.

870000.

870000.

869000. 869000.

868000. 868000.

867000. 867000.

260000. 265000. 270000. 275000.

Fig.6. Map of simulated values Zs of cadmium Cd content (the averaged picture on basis of 20-th con-

ditional simulations). The map of realizations standard deviation (Fig.7.) shows a somewhat different spatial distribution of

values δk than the ones shown by the point and block kriging maps of estimation standard deviation,

which is due to the different calculation procedures. The δk values are within a range of 1.2-11.7

mg/kg – similar to that obtained by point kriging (6.2-11.2 mg/kg), but higher than the one yielded by block kriging (2.6-9.3 mg/kg). X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter)

874000. 874000.

873000. 873000.

872000. 872000.

Y 871000. 871000.

870000. 870000. 869000. 869000.

868000. 868000.

867000. 867000.

260000. 265000. 270000. 275000. Fig.7. Map of standard deviation of realizations for cadmium Cd content (conditional simulation).

Inverse distances The spatial image of soil cadmium pollution obtained by the inverse distances technique shows a

much lower degree of soil Cd pollution (Fig.8.). The location of the pollution centre with its four foci

remains the same, but the Cd content values are underrated (4.3-19.6 mg/kg). The boundaries of the

polluted subareas are indistinct and contracted. The overall picture seems to be unreliable.



7

X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter) 874000. 874000.

873000. 873000.

872000. 872000.

Y 871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000. 867000.

260000. 265000. 270000. 275000.

Fig.8. Raster map of cadmium Cd content (the method of inverse distances).

Squared inverse distances This technique is a slight modification of the inverse distances method. The obtained map shows dis-

tinct (sharply outlined) areas with an anomalously high (over 35 mg/kg) and low (below 1.11 mg/kg)

soil Cd content (Fig.9.). The polluted subareas are more rounded and much smaller than the ones on

the (point and block) ordinary kriging maps. But the reliability of the extrapolated image in the un-

sampled areas seems to be doubtful. Summing up, this technique can turn out to be useful only in the

case of a densely and uniformly sampled area. X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter)

874000. 874000.

873000. 873000.

872000. 872000.

Y 871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000.

265000. 270000.

867000.

260000. 275000.

Fig.9. Raster map of cadmium Cd content (the method of squared inverse distances). Nearest neighbours The picture of soil Cd content spatial distribution obtained by the nearest neighbour technique has the

form of a geometric mosaic (Fig.10). Such a representation of soil pollution is not very convincing. It

seems that soil environment variation is highly complex and it does not lend itself to such a simple and

schematic description. The boundaries of Cd content areas with Z* values in a range of 1-42 mg/kg (Fig.10) coincide only

slightly with the maps obtained by means of (point and block) kriging, conditional simulation – turn-

ing bands and the squared-inverse-distances method. The latter technique can be used only for the

initial identification of a studied phenomenon and its results must be compared with those obtained by

other methods.



8

X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

Y (Meter) 874000. 874000.

873000. 873000.

872000. 872000.

871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000.

265000. 270000.

867000.

260000. 275000.

Fig.10. Raster map of cadmium Cd content (the method of nearest neighbors). Linear model of kriging The raster map produced by applying the linear model of kriging is the closest to the images obtained

by point and block kriging (Fig.2. and Fig.4.), especially by conditional simulation (Fig.6.). The spa-

tial distribution of Cd pollution in the form of one main centre made up of four foci (25-36 mg/kg) and

two smaller centres (13-16 mg/kg) differs only slightly from the one shown in (Fig.11.). The shape of

the aureolas which surround the centres and the rather indistinct zones of low soil Cd content are the

only pronounced differences. X (Meter)

260000. 265000. 270000. 275000.

877000. 877000.

876000. 876000.

875000. 875000.

(Meter

)

874000. 874000. 872000. 872000.

873000. 873000.

Y 871000. 871000.

870000. 870000.

869000. 869000.

868000. 868000.

867000.

265000. 270000.

867000.

260000. 275000.

Fig.11. Raster map of cadmium Cd content (the method of linear model kriging). RECAPITULATION All the interpolation techniques presented here, although differing considerably in their mathematical

assumptions as to the realization of computations and in their complexity and laboriousness, made it

possible to identify a centre of a high soil Cd content hazard in the Bukowno-Sławków area. The range

of average estimated values Z* differs between the techniques used: point kriging – 2.7-28.9 mg/kg,

block kriging – 2.9-27.4 mg/kg, linear model of kriging – 2.1-36.9 mg/kg, inverse distances method –

4.3-19.6 mg/kg, squared inverse distances method – 1.1-35.2 mg/kg, nearest neighbours method – 1-

42 mg/kg and conditional simulation (turning bands): 1.4-30.6 mg/kg.

A comparison of the ranges of values δk shows that block kriging gives the lowest values of estimation standard deviation (2.6-9.3 mg/kg) and for this reason this method is recommended for

environmental applications. Higher δk values were obtained by using turning bands conditional simulation (1.2-11.7 mg/kg) and point kriging (6.2-11.2 mg/kg).



9

The estimation methods and turning bands simulation (with Cd content variogram model parameters

taken into account) yield similar ranges of averages Z* or Zs. Among the interpolation techniques, the inverse distances technique greatly underestimates values Z*,

whereas the nearest neighbour technique overestimates them (1-42 mg/kg). The representations of soil

Cd content obtained by the other techniques are similar to each other. The centre of high Cd concentrations (visible on all the maps) is situated in the western part of the

studied area and it is associated with the operations of the Bolesław Mine&Metallurgical Plant (min-

ing and processing of zinc and lead ores) located there and with the increasing quantities of metallur-

gical wastes on the nearby dumps and in the sedimentation tanks (Fig.12.). It is highly probable that

besides the anthropogenic factors also the geology (natural factors) of the area has contributed signifi-

cantly to the contamination of the soils with cadmium [4]. The ore-bearing dolomite deposits lie near

the surface and they are rich not only in zinc and lead but also in cadmium. The historical, robber ex-

ploitation of the zinc and lead ore deposits and their processing (very primitive in comparison with

modern practice) which have been going on for centuries have left their stamp on the soil environment

in Bukowno and in the surrounding areas. 262.5 265.0 267.5 270.0 272.5 275.0

877.5 877.5

Pomorzany

875.0 Mine 875.0

Sławków Dumps

872.5 872.5

Bolesław M&M Plant

870.0

Bukowno

870.0 Town

867.5 867.5

262.5 265.0 267.5 270.0 272.5 275.0 Fig.12. Region of Bukowno – Sławków with the marked points of soil sampling

CONCLUSIONS The presented research methodology in which various interpolation techniques are used to identify

areas of high concentrations of the analyzed metal (cadmium) has proved to be a valuable tool for the

spatial description of phenomena occurring in an environment studied by geostatistical methods. Each

of the techniques has its advantage and disadvantages. The presented simple interpolation techniques seem to be useful only at the preliminary stage of pollu-

tion identification. The results obtained by means of them should be then verified by ordinary block

kriging estimation which because of its lower standard deviation values is often employed in environ-

mental studies. Better results were obtained by applying the kriging methods in conjunction with con-

ditional simulation – turning bands which, besides reflecting the fluctuations in the Cd content, also

brought out contamination centres invisible on the kriging maps. The estimated kriging surface is the only one, whereas the number of surfaces simulated for the same

data and statistical moments is infinite. Simulation is preferred over kriging in situations when the

model of the spatial variations in the real surface is more important than the estimation accuracy on the

local scale. The kriging methods smooth the actual variation of a studied phenomenon, whereas simu-

lation shows the true picture of the these changes [8].



10

One should choose a method which is most suitable for a given task and time in which it is to be car-

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