comparison of spatial prediction methods for mapping floodplain soil pollution

I i CATENA vol. 17, p. 535 550 Cremlingen 1990

C O M P A R I S O N OF SPATIAL P R E D I C T I O N M E T H O D S

FOR M A P P I N G F L O O D P L A I N SOIL P O L L U T I O N

H. Leenaers, J.R Okx, RA. Burrough Utrecht

Summary I Introduction

Due to past mining activities, the flood- The floodplain soils of the River Geul plain soils of the Geul river are polluted (The Netherlands) are polluted by heavy with heavy metals, particularly zinc, lead metals, i.e. lead, zinc and cadmium, and cadmium (LEENAERS et al. 1988, Several spatial prediction methods (lo- R A N G et al. 1986). The general pol- cal trend analysis, mathematical splines, lution pattern consists of a logarithmic inverse distance weighting, block kriging, decay with distance to the source of con- point kriging and point co-kriging) were taminants (LEENAERS et al. 1988), used to map zinc levels in a small and with local deviations that arc', caused by intensively sampled study area. Three variations of flood frequency and of sed- subsets of the set of 145 data points, con- imentary conditions during flood events. taining 12, 23 and 44 data respectively, The pollutants constrain the land use in were used to estimate zinc levels at 99 these areas, therefore detailed maps are test locations. Correlation coefficients of required that delineate zones with high linear relations between observed and es- concentration levels. Successful attempts timated zinc levels and contour maps of have been made by W O L F E N D E N & prediction errors indicate that weighted LEWIN (1977) and R A N G et al. (1987) local averaging methods perform better to relate the pollution level of floodplain than the other methods. In the study soils to floodplain characteristics such area, point co-kriging with elevation as as geomorphology, inundation frequency a co-variable outperforms all other meth- and soil type. These studies led to the ods when only few data on zinc are avail- production of choropleth maps that de-

able. lineated broad pollution zones, but did not provide detailed information about the continuous spatial variation of pollution levels within the zones.

ISSN 034l 8162 There are many point interpolation @1990 by CATENA VERLAG, D-3302 Cremlingen-Destedt, W. Germany techniques that can provide this type 0341-8162/90/5011851/US$ 2.00 + 0.25 o f information, see for example LAM

( ' A [ E N A A n lnterdiscipfinar~, J o u r n a l o f S O I L S C I E N C E H Y D R O L O G Y G E O M O R P H O I . O ( J 5

536 Leenaers, O k x & Burrough

(1983) and B U R R O U G H (1986). How- large heaps. Some of these heaps still ex- ever, to date only few attempts have ist. The presence of these point sources been made to compare spatial predic- has led to the dispersal of heavy met- tion methods quantitatively (BREGT et als through the catchment during and al. 1987, DAVIS 1976, LASLETT et al. after the active mining period. During 1987, K U I L E N B U R G et al. 1982). Ob- high flow stages flooding occurs and the viously, one seeks an interpolation tech- metal-rich suspended sediments are de- nique that gives the best results for a posited on the floodplains (LEENAERS given investment in observations. The et al. 1988). Due to erosion processes aim of this study is to evaluate some during periods of rainfall, the spoil heaps of these techniques (local trend analysis, are still active sources of heavy metals. mathematical splines, inverse distance Erosion of locally contaminated stream weighting, point kriging, block kriging banks is an additional source of heavy and point co-kriging) for the purpose of metals. mapping floodplain soil pollution. The

quality of the interpolation techniques 3 S p a t i a l prediction methods used is based on the correlation between

predicted and observed values at 99 test 3.1 Fitting of local trend surfaces locations and on the spatial distribution of prediction errors. The simplest way to describe gradual

variations of a property is to model them by polynomial regression. The idea is to

2 The study area fit a polynomial surface by least squares through the data points, where it is as- Significant occurrences of metal ore are sumed that the spatial coordinates X and found near Plombi6res and Kelmis, both Y are independent variables, and that Z, situated in the Belgian part of the Geul the property of interest, i,; the dependent basin (fig. 1). The exploitation of zinc

and lead ores in the Geul basin probably variable. In two dimensions the polyno- mials are surfaces of the form: began in the thirteenth century. Mining

activities were highest from 1820 to 1880. Z ( X , Y) = 2 (br~ " X ~ " Y~) During that period, maximum yearly ore

r + s < = p

production in the open-cast mines of Kelmis was about 135,000 tonnes, and where r and s are the polynomial coeffi- 1,300 men were employed in the zinc in- cients, p is the order of the trend surface dustries. The last mine closed in 1938, and b~ is a constant. A thorough ex- but the processing of metal ores contin- planation of this technique is given by ued until the 1950s. DAVIS (1986), RIPLEY (1981), CHOR-

Separation techniques exploiting the LEY & H A G ETT (1965) and WATSON differences in specific density were used (1971). Two important disadvantages of when mining the ores. These techniques this general trend analysis, i.e. the sus- were inefficient and resulted in high con- ceptibility to outliers and the inability centrations of ore particles and metal- to fit a low order polynomial through rich spoil in the effluent, which was dis- complex data (BURROUGH 1986), can charged directly into the river. The re- be reduced with a moving window ap- ject material and tailings were dumped in proach: only the data that are within a

( A T I N A Al~lnterdisciplinaryJournalc~[SOlI S ( I E N ( ' [ HYDROIfI(~Y (}FOMORPtI ( ) IO(IY

Spatial Prediction Methods £or Mapping Soil Poflution 537

,•[• t" " - "L. ..I J -,, 1 £ ~

.,. _ Valkenburg( , _ -..

I I._ ~--. Schin ~{o Geul

Gulpen •

/ f'e~len~ )L.~ i t FRG I

"~x+ # ÷ 't 'r ~ a" ~ , ~ u g g " e l ~ ÷ + "1"%~, ' - %

\ A Plombi~re~s ox-~ f - - / ~+ K

I ] .~ - I -+++ +./ . + + + ~ . ' ~ r" LUlUlW_~-/ "~r / Kelmis ~--~.~'~

\ / \ - % , ~, I

0 2 4 6 8 1Okra %,... ~..% ~ , / ; I I I I I

Fig. 1" The River Geul catchment.

specified search radius are used to esti- 3.2 Fitting of mathematical splines mate the proper ty of interest by fitting

a local trend surface. For our study we Given a set of datapoints , one of the fitted first order local trend surfaces by simplest mathemat ica l expressions for a using the nearest points within a search cont inuous surface that intersects these radius, which was derived f rom an anal- points is an interpolat ing polynomial o f ysis o f the spatial structure (see section the lowest order that passes through all on var iogram models), da tapoints (LAM 1983, R I P L E Y 1981,

D U B R U L E 1983, T I P P E R 1979). The major deficiency of this polynomial fit is that since the polynomial is entirely

(J , \ l ENA An Interdisciplinar~ Journal of S O I L S£ ' IENCE H Y D R O L O G k - G E O M O R P H O [ O G Y

538 Leenaers, Okx & Burrough

unconstrained, except at the datapoints, affected by uneven distribution of data the values attained between the points points since unequal weight will be as- may be drastically different from those signed to each of the points. Finally, at nearby data points (LAM 1983). This because this method is by definition a problem may be alleviated to a certain smoothing technique, maxima and min- extent by employing piecewise polyno- ima in the interpolated surface can oc- mial surfaces to cover the area. For our cur only at data points. The location study we used a local algorithm that ex- and magnitude of extreme values there- ploits only the four nearest data points fore, cannot be detected when they are to estimate Z at an unvisited site. not included as original sample points

(LAM 1983). However, the simplicity of 3.3 Inverse distance weighted the principle, the speed in calculation,

interpolation the ease of programming and reasonable results for many types of applications

The principle of distance weighting have led to wide adoption of this method, methods (WEAVER 1964, SHEPARD as well as various improvements (LAM 1968, McLAIN 1976) is to assign more 1983). weight to nearby points than to dis- tant points, which is based on the idea that observations located close together 3.4 Point kriging, block kriging and tend to be more alike than observations point co-kriging spaced further apart. The most common

Recent publications in soil science have form of the weighting function is the re-

demonstrated the application of re- ciprocal function 1/d 2, where d is the

gionalized variable theory to mapping distance between a nearby point and the soil variables (BURGESS & WEB- point to be estimated:

STER 1980a, 1980b, B U R R O U G H ~'/_lZ(xi) .do2 1986, DAVIS 1986, WEBSTER 1985).

n The theory provides a convenient means Z ( X j ) = Zi=I du 2 of summarizing soil spatial variability in

where the x; is the point at which the the form of a semi-variogram which can surface is to be interpolated. For d = 0 be used to estimate weights for interpo- the exact value of the original sampling lating the value of a given soil property point has to be preserved, otherwise the at an unsampled location. Kriging is a function goes to infinity. By changing form of weighted local averaging that the value of the inverse power parameter, is optimal in the sense that it provides the distribution of weights can be varied estimates of values at unsampled sites from being highly biased in favour of without bias and with minimum vari- near data points to giving nearly equal ance which is estimable. The weighting weight of all data points, procedure is similar to that used in in-

There are several disadvantages to verse distance weighted interpolation ex- weighting methods. First, the choice of a cept that the weights are derived from weighting function may introduce ambi- a geostatistical spatial analysis based on guity, especially when the characteristics the sample semivariogram. of the underlying surface are not known. Point kriging or punctual kriging is Second, the weighting method is easily an exact interpolator in the sense that

CAIENA An Interdisciplinary Journal o[ SOIL S ( I E N ( E IIYDROI f)GY (H OMt')RPHOI O(]~

Spatial Prediction Methods for Mapping Soil Pollution 539

Elevation ~ ~

?O'o~

Zinc M %

lOOK ~ ~ ~ ~

Fig. 2

a) Digital Elevation Model of the study area (x- and y-coordinates and elevation in m) b) Block diagram of Zn levels (mg/kg) in the study area.

I'¢.1 L N A An I n t e r d i s d p l i n a r ? J o u r n a l o f S O I L SCIENC.'E I~ IYDROLO(3Y G E O M O R P H O L O ( ~


at sampled locations, interpolated and at 145 locations. Elevation relative to measured values coincide. Point kriging mean sea level was determined for each implies that all interpolated values relate sample location by using a level. All to an area or volume that is equivalent to samples were dried for 24 hours at 60°C, the area or volume of an original sample, and crushed with a mortar. Two gram Given the often large, short-range nature of this material was gently boiled for of soil variation, point kriging sometimes 2 hours with 20 ml 30'}/0 HNO3. The results in maps that have many sharp extract was then separated from the sed- spikes or pits at the datapoints. If av- iment by centrifugation and brought to erage values of soil properties over ar- 40 ml with distilled water. The concen- eas are considered more interesting than trations of zinc was determined by direct values at points, block kriging is the al- flame absorption spectometry. ternative (BURGESS et al. 1980b). The Block diagrams of elevation and zinc estimation variances obtained for block levels are shown in fig. 2a and b. It krigingare usually substantially less than is clear that zinc levels are inversely those for point kriging. The results are related to elevation, which can be ex- much smoother surfaces free from the plained by the fact that low-lying ar- pits and spikes resulting from point krig- eas have a higher risk of inundation. It ing. will be shown that benefit may be drawn

Co-kriging is a logical extension of from the relation between zinc levels and kriging to situations where two or elevation by means of a co-kriging pro- more variables are spatially interdepen- cedure. dent and the one of immediate interest is undersampled ( JOURNAL & 4.2 The test procedure H U I J B R E G T S 1978, McBRATNEY & WEBSTER 1983, MYERS 1982, 1984, The set of 145 observations on topsoil LEENAERS et al. 1989). Co-kriging zinc levels was divided into two subsets. may be a useful method for interpolating The larger set contained 99 data points a property that is expensive to measure (and one missing value) and was set aside by making use of a spatial correlation for future validation. The smaller set between the property of interest and an of 44 data (and one missing value) was attribute that is less expensive to mea- used for the estimation of experimental sure. semi-variograms of zinc levels and eleva-

tion and a cross semi-variogram of both.

4 Experimental and analytical The same set was used to design three methods configurations of sample locations with

different densities, containing 12, 23 and 44 observations respectively (fig. 3). 4.1 Sampling scheme and laboratory

analyses For each configuration, zinc levels were estimated using the interpolation

A regular grid of 5x29 (grid size 5x5 m) methods discussed above for locations was laid out in the floodplain area of the in the original sampling grid. For co- River Geul, near the village of Valken- kriging, the data on zinc levels were sup- burg (fig. 1); 100 g-samples of top soil plemented by all elevation data (n=145). material (depth: 0~10 cm) were collected Correlation coefficients of linear reid-

( ' A ' I E N A An In te rd i sc ip l ina ry J o u r n a l n l S O l [ S( [ E N ( | H Y I ) R O I , ( ) G ' { ( } E O M l l R P H O I O ( i Y


1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 I 0 1 2 0 1 3 0 1 4 0

20 , -~ , , , ~ , ~ , , , , " ~ "

~o • * ® (~) ® * ® <~) • * ® ~ ) • 0

I -,-'L | ~ , ~L I ~ [ -,-J~ , ~ I ~L I ~ I -,-'L , ~ [ T"L I ~ I -l-'L |

10 20 30 4-0 50 60 70 80 90 1 O0 110 120 130 140

set of 12 samples

O set of 23 samples

set of 44 samples

Fig. 3: Sampling grids in the study area.

attribute(s) Co C C0+C a a'

Zn 4588 106694 111282 17.1 29.6 elevation 417 2657 3074 23.8 40.8 Zn-elevation -697 -15992 -16689 20.7 35.8

Co: nugget; C0+C: sill, a': practical range.

Tab. 1 : Parameters o.1 the theoretical Gaussian variogram Jhnctions.

t ions between es t imated and observed tances much less than the sampl ing in- zinc levels were used to evaluate the pre- terval ( O L I V E R & W E B S T E R 1986). cision o f the es t imates and to compare In most instances, the semivar iance in- the pred ic t ion methods . In addi t ion, con- creases from the nugget var iance to a tour maps o f the abso lu te pred ic t ion er- max imum level beyond which it remains rots were p roduced in o rder to judge level. Such semi-var iograms are transi- their spat ial d is t r ibut ion, tive, the ma x imum value is known as the

sill, which is the a pr ior i var iance o f the variable. The lag dis tance at which the

4.3 Estimation of the semi-variogram sill is reached is known as the range. models

As an observat ion of a var iable can In general , a few simple features con- depend in the stat is t ical sense on obser-

t r ibute to the form of a semi-var iogram, vat ions of the same var iable at o ther lo- The semivar iance at zero lag must be cat ions nearby, they can also be spat ia l ly zero, but in pract ice the ex t rapo la t ed related to observat ions o f o ther vari- semivar iogram usual ly intercepts the or- ables. Such var iables are co-regional ized. d ina te at a posi t ive value known as Ana logous to the single variable, the the 'nugget variance' . The nugget vari- dependence between two var iables can ance can arise from measurement error, be expressed by a cross semi-var iogram discrete r a n d o m var ia t ion and spat ia l ly ( M c B R A T N E Y & W E B S T E R 1983). In dependen t var ia t ion occurr ing over dis- this study, e levat ion was chosen as a co-

('ATIINA An Interdisciplinary Journal of SOIL S('IEN(!E HYDROI O(}Y GEOMORPHOI ()Gk


7(h)

150000 8

o

D

1 0 0 0 0 0 ~ D C

50000 /

0 i 1 ~ I i I i I I 0 10 20 30 40 50

tag,h (m)

7(h)

0 b

-5000

-10000

-150OO

-20000 I I I I I i I I I 0 10 20 30 40 50

IGg,h (m)

Fig. 4

a) semi-variogram for Zn (semi-variance in mg/kg 2) b) cross-variogram for Zn and elevation (cross-variance in m.mg/kg).

CATENA An Interdisciplina D .Iournal of SOl [ ~;(][gNCE HYDROLC)(]Y ( ] [ iOMORPIIOLO(; ' r

Spatial Prediction Methods For Mapping Soil Pollution 543

number of input data spatial prediction method n- 12 n=23 n=44

first order local trend surfaces 0.41 0.57 0.63 mathematical splines 0.10 0.61 0.77 inverse distance weighting 0.77 0.75 0.83 block kriging 0.73 0.79 0.84 point kriging 0.76 0.80 0.90 point co-kriging 0.85 0.89 0.89

Tab. 2: Correlation coefficients o['linear relations between estimated and observed Zn levels (n=99) for sets of input data.

variable for estimating zinc levels. 712(h) < = V/(Tl(h) * 72(h)) for all h > = 0 Most samples were located in the natu-

ral levee area where only small variations where ] "12 - - c r o s s s e m i - v a r i a n c e ;

in elevation existed; 19 observations were ~,~ - auto semi-variance zinc; in a abandoned channel loop. )'2 - auto semi-variance elevation,

Multidirectional semi-variograms for was checked to guarantee a positive cok- zinc (fig. 4a) and elevation were corn- riging variance under all circumstances puted for the set o f 44 data. Due to ( J O U R N E L & H U I J B R E G T S 1978, the shape of the 5x29 grid, the results M Y E R S 1982, MYERS, 1984). The are dominated by effects of the spatial parameter o f the fitted semi-variogram variation parallel to the longer axis. The models are listed in tab. 1. During the parameters required for kriging were ob- block kriging procedure a block size of tained by fitting a gaussian equat ion: 20×20 m was used.

For the other prediction methods, the 7(h 'C° 'C 'a )=C°+C(1-exp ( -h2 /a2 ) ' parameters were set in such a man-

where ner that for each estimate only data 7 semi-variance; fall within the range (a') o f the semi- h = lag (m); var iogram of zinc are used: for inverse Co nugget variance; distance weighting and the local trend co + c - sill; method the search radius was set equal to a = d i s t a n c e p a r a m e t e r , the range (a') and for the fitting o f math- through the estimates o f experimental ematical splines the number' o f nearest semivariance. The practical range (a') points was set to 4. was determined to be the lag at which 95% of the sill is reached. The cross semi-variogram for topsoil zinc and tel- 5 Results and discussion ative elevation (fig. 4b) could also be

5.1 Goodness-of-fit of the modelled modelled adequately using a gaussian zinc content surfaces model. The cross semi-variances are neg- ative because of the inverse relationship Fig. 5 shows the contours of predicted between the variables. In the process of zinc levels based on interpolation from cross semi-variogram fitting the Cauchy- 44 data points. As might be expected Schwarz inequality: from its approximate nature, the fitting

( ' , " , l E N A A n Interdisciphnar '~ J o u r n a l o f S O I L S C I E N C E H Y D R O L O G Y G E O M ( J R P H O I O G Y


of first order local trend surfaces yields tively. For block kriging, this can be a rather smooth zinc surface, whose gen- explained by the fact that this method eralized nature is most striking in the smoothesextreme values over a 2 0 x 2 0 m vicinity of the abandoned channel where area. The relatively small improvements steep gradients should prevail (fig. 2). Vi- by inverse distance weighting when the sual examination of the contours suggest number of data points was increased that by fitting mathematical splines and however, are probably due to the choice by local averaging methods (inverse dis- of the weighting function. Since this tance weighting and kr iging)more accu- function does not relate to the spatial rate results are obtained, correlation structure of the data, its ap-

Correlation coefficients of the linear plicability shows an undesirable depen- relations between observed and predicted dence on the input data density and/or zinc levels were used as a measure of data geometry. The fact that the cor- goodness-of-fit (tab. 2). relation coefficient decreased from 0.77

For all methods the correlation coeffi- to 0.75 when the number of input data cients increases with an increasing num- points was increased from 12 to 23 illus- ber of input data. The rate of increase trates this behaviour. however, is notably different. The in-

crease for the local trend method oc- 5.2 The spatial distribution of curs gradually (from r=0.41 for n=12 to estimation errors r=0.63 for n=44), while the performance of the splines method changes from a For environmental assessment studies very poor (r=0.10 for n=12) to a rela- not only the goodness-of-fit of the spa- tively high level (r=0.77 for n=44). This tial model but also some idea about the pattern illustrates the different qualities magnitude of the prediction errors is of of smoothing, approximate methods and importance. Instead of using measures flexible, exact prediction methods, of central tendency and dispersion, we

When only 12 data points are avail- decided to produce contour maps of pre- able, these two methods are outper- diction errors. For each interpolation formed by the weighted local averaging technique, the grids containing the esti- methods (inverse distance weighting and mates were subtracted from the grid con- kriging). The correlation coefficient is taining the 145 data at the data points, remarkably high for co-kriging, which and the absolute value of the difference illustrates the advantage of using addi- at each grid intersection was computed. tional, spatially correlated, information The resulting grids have the same dimen- when only few data points are available sions as the sampling grid and contour on the variable of interest. The rela- maps could be produced. As for the ap- tire benefit of co-kriging decreases as proximate methods, errors that occur at the number of data points with known the input data locations are included in zinc levels increases: for n=44 point krig- the maps. ing and point co-kriging perform equally By this means, not only the magnitude well (r=0.90 and r=0.89 resp.). With but also the spatial distribution of pre- these data points, the correlation coeffi- diction errors can be studied. The occur- cients for inverse distance weighting and rence of larger errors should preferably block kriging are 0.83 and 0.84 respec- be restricted to small, well-defined areas.

C A T E N A An In te rd i sc ip l ina ry J o u r n a l o1" S O I l N ( ' I E N ( ' E H Y D R O I O ( ~ Y ( H O M O R P H O [ O ( ~ ' I


o ~o 20 ~o ~o ~o ~o 70 ~o 90 ,oo ~io ,~o ~o ~o

20 . . . . i , t , , / . . . . . ~ t ,,~,/ , ~ , ,~ t \ , , , \ ~ , i 2 °

~o - ~ o o ~ , ~ o ~o

0 I I I I i i i i i i I i i i i \ i % ~ i i l / i 0

,o \/%F~-__/~ o ° J,o

0 I I I /~ I I I I i'rx i i I I I I I i I o

.o 7 . . . . . . . . . . .._,...W__t_ yt ' t '77t , .~ " ~ .I '° "L.._.~ -.-y ,,~ 7/ ~ / . . . . . . . ' ' " '

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10 10

o , , i . - I - - - I . , , , , , / T " , , \ \ ~ I \ ' ' ' o

~ . . ' Y , , ~ - - Y - '~ ' I ~' \ t' ' ' 20

1o ~ - - - "~J 1o

0 m ~ J , I I i i .¢'1'~'~ i ~ I I i i \ ' I I 0

P ? ° ._ o , , , , ~ , , , , , , , I / , I ~ 7 . , \ " , , , \ \ ' ~ , \ , , 0 10 2 0 3 0 4.0 5 0 6 0 7 0 8 0 9 0 1 O0 1 1 0 1 2 0 130 140

Fig. 5: Contours of Zn levels (mg/kg) obtained from 44 data by a).first order local trend analysis; b) mathematical splines; c) inverse distance weighting; d) block kriging; e) point kriging; f ) point co-kriging; and g) contours of Zn levels based on all 143 data..


0 10 20 50 40 50 60 7(3 80 90 100 110 120 1,30 140 20 20

10 10

0 0

20 20

100 100

ioo 10

0

20 ~ 20

o :::::::::::::::::::::::::::::::: .............. ~ o

0 0

'° ........ !i i i,i ~ i ~i:ii 1o'0 20 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ..:~::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ....: : 20

o K : o • :::::::::::::::::::::: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :::::::::::::: ::::::::::::::

o ........... ~::::::'::::: ::::::::::::::::::::::: o 0 ~0 20 30 40 ~0 60 70 80 90 ~00 ~0 ~20 ~ 0 ~,,0

o - l o o r~o /kg

~ 1 0 0 - - 3 0 0 m g / k g

i ~ 300- 500 rag/kg m > 500 mg/kg

Fig. 6: Contours of prediction errors (mg/kg Zn) by a)f irs t order local trend analysis," b) mathematical shines; c) inverse distance weighting; d) block kriging ; e) point kriging; f ) point co-kriging..

( A I ENA An [nterd sciplinary Journ ol SOIl. S('IEN('E HYDROI O(;Y (;E(IMORI'IIOLO(IY


However, one should be aware of the fact as the basis for this study shows large that approximate methods (block krig- short range variations (fig. 2). It is clear ing and fitting of local trend surfaces), that approximate methods, that do not due to their nature of generalization, are aim at preserving the original data nor likely to produce a pattern of regularly at exact prediction at unvisited sites, will distributed errors, yield a smooth zinc surface that ignores

Fig. 6 shows contour maps of abso- extreme values. Therefore, fitting of local lute errors (>100 mg/kg) that were ob- trend surfaces yields poor results and is tained by interpolation from 44 input not recommended for this type of data. data points. With an exception for block The main purpose of using this method kriging, all methods yield large errors was to illustrate the difference with more only in those areas where high zinc lev- appropriate methods. els prevail. The errors produced by fitting Fitting of mathematical splines, con- local trend surfaces are highest and cover trary to trend surfaces, has the advan- a relatively large area. The error maps tage that very steep gradients can easily obtained by inverse distance weighting be modelled. However, it lacks the pos- and fitting of mathematical splines are sibility to attach more weight to nearby much alike: the area that is covered points. By inverse distance weighting, by the occurrence of errors >100 mg/kg the problem of attaching equal weights are about equally large and in the aban- to all data within the search radius is doned channel a number (3M) of errors to a certain extent alleviated. However, >500 mg/kg occur. For block kriging the since the weights are derived from some occurrence of errors >500 mg/kg is re- convenient function and do not relate to stricted to one grid cell only, but smaller the spatial correlation structure of the errors of 100-300 mg/kg can be observed data, the results depend on the often in the entire area. Point kriging and unknown applicability of the chosen point co-kriging combine both advan- function. In this study the function l id 2 tages: a relatively small area is covered provided excellent results with 12 data by errors in the ranges 100 300 and 300- points, where the search radius was set 500 mg/kg and in only one grid cell an equal to the range of the semi-variogram error of >500 mg/kg occurred, of zinc. However, compared to point

kriging relatively small improvements oc-

5.3 Diseussion curred when the number of data points was enlarged, which illustrates the inabil-

The performance of spatial prediction ity of this function to optimally exploit methods will likely depend on the type additional information. By choosing a of data that are used and on the spa- few other weighting functions we found tial correlation structure of the data. It that with the set of 44 data points weight- is therefore impossible to draw general ing by 1/d 4 was sligthly more appropri- conclusions from the results that were ate: the correlation between predicted obtained in this case study. Nevertheless, and observed values increased from 0.83 this study illustrates some of the quali- to 0.86. It is clear that in practice one will ties of the methods, as will be discussed not always have the possibility to check below, the validity of the weighting function.

The surface ofzinclevels that was used The semi-variogram model of zinc

( NI t N A An In te rd i sc ip l ina ry J o u r n a l o f S O I L S C I E N C E H Y D R O [ . O G Y G E O M O R P H O L O G Y


provide weights for interpolation by timates of zinc levels when many data point kriging and block kriging. The points are available in areas with large advantage of this sample-based weight- short range variations. ing function is best illustrated by the Because of their capability to ac- results of point kriging. Both in terms count for spatial dependence, local of goodness-of-fit and the spatial distri- weighted averaging methods (inverse dis- bution of prediction errors this method tance weighting and kriging) require less performs better than any of the other data to provide an acceptable estimate methods that do not use additional el- of the spatial distribution of zinc levels. evation data. The smoothing nature of However, in this case study the geosta- block kriging is nicely illustrated by the tistically based sample semi-variogram fact that the area covered by small errors provides better weights for interpolat- (10(~300 mg/kg)is relatively large, ing zinc levels than inverse distance

The only method that clearly shows weighting functions. Major improve- major advantages over all other methods ment in the spatial prediction of" zinc is point co-kriging. Its capability to bene- levels in the Geul floodplains is pro- fit from inexpensive additional informa- vided by point co-kriging with elevation tion by making use of a strong spatial as a co-variable. Inexpensive elevation correlation between zinc levels and ele- data exhibit a strong spatial correlation w~tion, accounts for the fact that with with zinc and contain information on 12 input data only an adequate spatial the dispersal mechanism of zinc. The re- model of zinc levels could be obtained, sults presented demonstrate that the co- Taking advantage of this spatial corre- kriging method can be greatly improved lation during the design of a sampling when there is a physical understanding strategy, a substantial reduction of costs of the contamination process. Moreover, can be realized. In this study for exam- without loss of precision a cost reduc- ple, the cost for sampling and laboratory tion of 30% could be realized because analysis is c. ECU 25,- per sample, and co-kriging requires less data points. the collection of elevation data costs c.

ECU 1,30 per location. Thus, if one Acknowledgements neglects the difference in required computer time, for point kriging with 44 data This study was financed by the Nether- on zinc an amount of c. ECU 1150,- lands Organization for Scientific Re- is needed, while point co-kriging with search (N.E.O.). We wish to thank R 145 elevation data and 23 data on zinc Nienhuis (Free University of Amster- costs ECU 790,-. Without loss of preci- dam) and J. van Keulen for geostatistical sion --- the two methods yield the same advice and software support. goodness-of-fit (tab. 2) - - a cost reduction of more than 30% could be brought References about.

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( 'A ' IENA An Interdisciplinary Journal o l S O I l SCIEN{'L: H Y D R { } L ( , ( ; Y ( } h O M O R I ' H ( I L ( ) G Y


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Addresses of authors: H. Leenaers J.P. Okx* P.A. Burrough Department of Physical Geography University of Utrecht p.o. Boxc 80.115 3508 TC Utrecht The Netherlands * present address: TAUW Infra Consult B.V. P.O. Box 479 7400 AL Deventer The Netherlands

CAT[NA An Interdisciplinary Journal of SOil SCI[NCE HYDROI.O(IY ( ]EOMORPIIOIO( ;Y

comparison of spatial prediction methods for mapping floodplain soil pollution

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