Chemosphere, Vol.20, Nos.lO-12, pp 1567-1573, 1990 0045-6535/90 $3.00 + .O0 Printed in Great Britain Pergamon Press plc

AN ATTEMPT TO PROVIDE A FRACTAL MODEL FOR THE DESCRIPTION OF THE TCDD DISTRIBUTION IN ALL THE TERRITORY AROUND SEVESO

(MILAN, ITALY)

G. Belli(1,3), G. Bressi(1,2), L. Carrioli(5), S. Cerlesi(1,4), M. Diani(6), S. P. Ratti(1,2), G. Salvadori(1)

(1)Dipartimento di Fisica Nucleare e Teorica, Universit& di Pavia; (2)lstituto Nazionale di Fisica Nucleare, Sezione di Pavia; (3)lstituto Tecnico lndustriale "G.Cardano", Pavia; (4)Regione Lombardia, Milano; (5)lstituto di Analisi Numerica - C.N.R., Pavia; (6)Dipartimento di Informatica e Sist. - Facolt& di Ingegneria - Pavia.

1. INTRODUCTION

In this paper we present a Random Fractal Model to describe the Tetra-Chloro- DiBenzo-Dioxin (TCDD) pollution on the ground due to the accident occurred on July 10 th 1976, when the explosion of a chemical reactor spread a large amount of the toxic substance on about 20000 hectares around the lcmesa Factory in the Seveso township.

The area of the accident was constantly surveyed for over 10 years[I]; in this paper only the data from the systematic investigations, performed according to rational regular grids during the years 1980-81, have been used.

Previous studies reached an empirical description of the TCDD distribution in Zone "A" using Tchebichev Polynomials as approximants and Shepard Method to enrich the data samples and to show a tridimensional view of the TCDD spread in that Zone[2a,b]. In all the Zone "A+B+R" (where the density of sampling was very low: ratio between sampled and total area in Zone "A" is about a factor 10 more than the same ratio in Zone "A+B+R"[3]), making use of only 10 free parameters (applied to various relations between TCDD values and polar coordinates), an Empirical Method[ 4] was tested mostly for epidemiological purposes, able to give numerical responses also in locations where no experimental measurements were made.

Although the empirical method was "good enough" for its use in epidemiological studies[5,6], we see the possibility of a more reliable description of the TCDD distribution on a large area around Seveso taking advantage of the powerfulness of the relatively new fractal algorithm in recovering the information data and in decanting for the unavailable fluctuations which are intrinsic to the phenomenon.

2 NATURAL PHENOMENA AND FRACTAL THEORY

A fractal figure has an unsmooth shape scaling structure; the shape can be "self- similar", that is it appears unchanged when examined by varying magnification.

Fractals are sets whose Hausdorff-Besicovitch dimension (which can be a non- integer number that measures the degree of "disconnectedness" or "contortion" or even the capacity of filling the space of a set) is greater than the topological one[Ta,b,c].

We find fractal structures almost everywhere in nature[7a,d]: in the geometry of rivers and coastlines, in biological structures such as trees and bronchioli, in the

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organization of the matter in the Universe, in the structure of the turbulence and so on.

A phenomenon neither continuous in space nor in time may have, mathematically speaking, "random fractal properties"; using fractal techniques to study a phenomenon showing fractal properties one can take into account a series of "non-benign" random fluctuations[8a].

Several observations[ 8a,b,c] of clouds shape and rain-fall fields structures demonstrate that the "self-similarity" and the "scaling" properties hold, so that the fractal nature of the phenomenon turns out.

On the basis of the similarity in the dynamical evolution of rainfall[ 8a] and toxic matter deposition, we assume that the latter phenomenon has a fractal behavior, so that the "geometrical" distribution of the dioxin on the ground around Seveso should be of fractal type.

3. CALCULATION OF THE FRACTAL DIMENSION

The algorithm used for the calculation of the fractal dimension D* of any tridimensional set[ 9a,b,c] comes from the relation between D* and the boundary area A measured by the iterative investigation of such sets with a sphere of radius p[6d]:

A (p) : Kp (2-D*) ( 1 ) where K is a constant. The measure of the area A (p) is given according to the following steps: a) the function f(x,y), which defines the fractal surface in the space, is

"approximated from the top" by means of the function fC(x,y) _> f(x,y). This function is calculated by smoothing f(x,y) with a mathematical procedure called "closing"[8d]:

fC(x,y) = C[ f (x ,y ) ,B(p) ] ( 2 )

where B(p) is the sphere of radius p; b) similarly, f(x,y) is "approximated from the bottom" by means of the

function fO(x,y) _< f(x,y) using a mathematical procedure called "opening"[gd]: fO(x,y) = O[ f (x ,y) ,B(p) ] ( 3 )

c) A(p) is evaluated using the equation:

A(p) = V(p)/2p ( 4 )

where V(p) is the volume enclosed between the approximant from the top and from the bottom of f(x,y), that is:

V(p) = j'j" [fC(x,y) - fO(x,y)] dx dy ( 5 )

By changing the value of p, the iterative procedure a)-b)-c) gives a measure of

A(p) related to p by (1). Using the logarithmic values, the equation (1) becomes:

log [A(p)] = (2-D*) log(p) + log(K) ( 6 )

In the plane Iog(p)-Iog[A(p)] the equation (6) is a line of slope (2-D'). The method described above, applied to the analytical measurements of TCDD

collected in Zone "A" and in Zone "A+B+R", gives a set of experimental values distributed on a straight line; interpolating such values, the slope of the line fitted by the least square method allows the calculation of the fraclal dimension D* of the "geometrical" distribution of the data.

When the radius p is sufficiently great, figures la and lb show a very good linear behavior and we find, respectively, the values D'= 2.692481 in Zone "A" and D*= 2.694976 in Zone "A+B+R".

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l=

m "O

e -

CALCULATION OF THE FRACTAL DIMENSION IN ZONE "A"

-1,4

-1,6 -

-1,8 -

-2,0 -

-2,2 -

-2,4 -

Log (radius appr. sph. )

-2,6 , , , , * ,0 1,5 2,0 2,5 3,0 3,5

FractalD = 2.69248Dimensi°n: I

Fig. la - Calculation of the fractal dimension of the data distribution in Zone "A".

CALCULATION OF THE FRACTAL DIMENSION IN ZONE "ABR"

-2,0

~ -2,2

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~ -2,8 -3,0

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Fig• Ib - Calculation of the fractal dimension of the data distribution in Zone "A+B+R"•

Since D" turns out to be nearly the same both in Zone "A" and in Zone "A+B+R", we have an experimental confirmation the investigated phenomenon is "self-similar" and "scaling".

4. THE FRACTAL MODEL

Our Fractal Model is based on the Theory of the Fractal Sum of Pulses[8b]: it states that the dynamical evolution of a phenomenon in a place can be recovered by means of a properly distributed series of randomly overlapped Primary Pulses•

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The set of experimental measurements constitutes the base to built the centers of the primary pulses needed to generate "fractal distributed" numerical values, which represents an enrichment of the original set of data.

The Primary Pulses mentioned above have the shape of "bubble" (fig. 2a) and each of them bears a random contribute to the simulation (fig. 2b). The algorithm works[10b] as follows:

1 ) using the log values of the measurements as input data-set for the model, we create a starting matrix Ms whose topographical extension corresponds to the polluted area and whose subdivision grid is a parameter to be properly calibrated and related to the sampling grid. All the data topographically belonging to a single cell gives a mean value which fills in the matrix M s (see an example of 1980-81 data set in zone "A+B+R" in fig. 3a).

A bidimensional "probability" matrix P proportional to the mean value of TCDD for each cell of Ms is built: a TCDD peak-value in a cell of Ms is translated (by means of P) into a sharp rise of the probability to choose that cell;

2 ) a set of coordinates Ci=(xi,Yi) is randomly generated[ 9a] according to the probability distribution given by P; each pair (xi,Yi) is the center of a pulse Pi and we see the number of generated pulses is proportional to the local values of pollutant. Therefore, in a random extraction, the centers Prs are concentrated around the peak values.

3 ) each pulse Pi covers a circular area Ai (centered in Ci) whose extension is given by the hyperbolic distribution law

Pr (Ai> a ) = a °1 ( 8 ) and the intensity li of Pi is related to Ai by means of:

l i= -+Ai 1/D ( 9 ) where D is the fraetal dimension of the phenomenon. The generation of the extension of the areas is not set completely free in order to

obtain a more realistic simulation. Given a point Ci, the maximum generable value of the area Ai (centered in Ci) depends upon the distance Ri of Ci from the Icmesa Factory and is chosen in the reverse order with Ri. We checked[ 1°hI that such a correction does not affect much the hyperbolicity of the generated areas distribution (8).

pR,I./.ARY - PULS[S Mop-of -Zone - A - with - BUBBLES

a) b)

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Ol I 9~

o I OJ S.25 O~l I 10J 11

O.A ¢4 $,7S 104 I01

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Fig. 2 - Example of Primary Pulses ("bubble"); b) Example of generation of "bubbles" in Zone "A".

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4) the algorithm stores the contribute given by each pulse Piin a final histogram Hf which topographically is equal to Ms but its subdivision grid turns out to be thinner and thus provides a fractal enrichment of the data-set.

Once ended the generation of the pulses, Hf represents the fractal description of the phenomenon in all the territory covered by the topographical boundaries of Hf; then, only the data enclosed in a chosen area (e. g. Zone "A" or Zone "A+B+R") are taken. The final description of the TCDD distribution is obtained by renormalizing the values of Hf to the range [TCDDmi n , TCDDmax] in the selected zone. In other words, we assume that Hf shows the fractal fluctuations of the data.

5. ANALYSIS OF THE RESULTS iN ZONE "A, B, R"

Starting from the data-set of the 1980-81 Campaign, the input distribution shown in figure 3a is generated; the Model succeeds in the fractal enrichment of the original data-set and the result (after the application of a smoothing algorithm) is presented in figure 3b: it demonstrates the model is more than adequate to describe the phenomenon since it is able, at least, to reproduce the trend of the original data.

Each cell of the final histogram Hf contains a value which is the average of the fluctuations related to a possible thinner grid subdivision of the cell itself, according to the initial assumption of a self-similar and scaling distribution of the TCDD on the ground.

The Model fills all the gaps enclosed in the contour of Zone "A, B, R" in a reliable way, giving a reasonable description of the pollution also in locations where no measurements were taken.

The comparison between the original data-set and the "fractally" enriched one is performed using the histograms of the values Iog(TCDD) and Iog(FRACTAL prediction) versus the coordinates X and Y: the results are shown in fig.s 4a and 4b.

The simulation shows the maintenance of the characteristics of the analytical measures where the sampling has been done.

S ~ . S O - (It o l y ) - - - F~ACTAL- Simulot~On ~ ,L~#ESO-Oto l y ) - - -ORI (~ INAL-Do toSe~- ( m e o n - volue~

a)

4

b)

Fig. 3 a) Lego-plot of the distribution of the original data (1980-81 Campaign - mean values); b) Output of the Fractal Model.

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Fig. 4 - Compar ison be tween the or ig inal and the "fractal ly" en r i ched data- sets along: a) X direction and b) Y direction.

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6. CONCLUSIONS

We are more than satisfied for the fact that, after ten years of continuous studies on the Seveso accident, we succeeded in providing a Global Description of the TCDD distribution on soil in all the area. However the work on the fractal model applied to the Seveso Case is still in progress. It must be stressed here that, in order to give the final description of the phenomenon, we still have to properly connect measurements performed by different Laboratories in different years to account for the analytical biases[2].

References

[1] S.P.Ratti, G.Belli, S.Cerlesi (1987) : Mathematical Approach to Data Analysis in Environmental Science - The lecture of Seveso. Chemosphere 18 pp.855-860;

[2a] G. Belli, S. Cerlesi, E. Milani, S. P. Ratti: Journal of Toxical and Environmental Chemistry, vol. 22, Gordon and Breach, Science Publishers, Inc. U.K, pp.101-130, 1989;

[2b] S. P. Ratti, G. Belli, A. Lanza, S, Ceflesi in: Rappe, Choudhary, Keith: Chlorinate Dioxins and Dibenzofurans in Perspective, Ed. Lewis Publ., Inc., pp.467-476, 1986;

[3] S. P. Ratti, G. Belli, G. Bressi, S. Cerlesi, G. Zocchetti: Proc. of "World Conference Chemical Accidents" Ed. CEP, Consultant Ltd., Edinburgh, U.K., pp.373-376, 1 987;

[4] S. Cerlesi, A. Di Domenico, S. P. Ratti: "A two-component model to describe 2,3,7,8 tetraclorodibenzodioxin (TCDD) vanishing patterns in the Seveso (Milan,Italy) soil" (presented to this conference);

[5] S.P. Ratti, G.Belli, G.Bressi, S.Cerlesi, C.Zocchetti: Chemosphere 18 Nos 1-6, pp.921-924, 1989;

[6] P.A.Bertazzi, C.Zocchetti, A.C.Pastori, S.Guercilena, M.Sanarico, L.Radice: American journal of Epidemiology' vo1.129, n.6, pp.1187-1200, 1989;

[7a] B. B. Mandelbrot: "Gli Oggetti Frattali"; Ed. Einaudi, Torino Italy,1987; [7b] K. J. Falconer: "The Geometry of Fractal Sets"; Ed. Cambridge University Press,

Cambridge U.K., 1988; [7c] V. Checcucci, A. Tognoli, E. Vesentini: "Lezioni di Topologia Generale"; Ed.

Feltrinelli, Milano, Italy, 1977; [7d] B. 8. Mandelbrot: "The Fractal Geometry of Nature"; Ed. Freeman, San Francisco

U.S.A., 1982; [8a] S. Lovejoy: "The Statistical Characterization of Rain Areas in terms of

Fractals', Conference on Radar Meteorology, Toronto, Canada, 1981; [8b] S. Lovejoy, B. B. Mandelbrot: Tellus, vol. 37.4, pp.209-232 , 1985; [8c] S. Lovejoy: Science, vol. 216, pp.185-187, 1982; [9a] G.Matheron: "Random Sets and Integral Geometry", Wiley, New York, 1975; [9b] S.R. Sternberg: "Grayscale Morphology", Computer Vision, Graphics and Image

Processing, vol.35, pp.333-355, 1986; [9c] D.D. Giusto, E. Parodi, G. Risaliti, G.Vernazza: "SAR Image Processing for

Classification Purposes", proc. of SPIE, vol. 1027, Amburgo, pp.214-221, 1988; [gd] j . Serra: "Image Analysis and Mathematical Morphology", Academic Press, New

York, 1982; [10a] R. Brun, D. Lienart: "HBOOK", Cern Computer Centre Program Library Y250,

Geneve (Svizzera), October 1987; [1 0b] F.J. Roos: "Function Minimization and Error Analysis", Cern Computer Centre

Program Library D506 (GENLI8), Geneve (Svizzera), May 1967; [11] S. P. Ratti, G. Belli, G. Bressi, S. Cerlesi, E. Calligarich in: O. Hutzinger,

R.W.Frei, E.Merian, F.Pocchiari: Chlorinated Dioxins and Related Compounds, Pergamon Press Oxford and New York, vol.5, pp.155-172, 1982.