The results yield the following considerations and preliminary conclusions.

First of all, data may contain much more information that initially expected. Moreover, data need some preliminary quality assurance, since the selection of reliable samples is indeed fundamental in order to provide certified and consistent measurements as a starting point. Unfortunately, due to the difficulty in accessing the data, we were not in position of doing the same for Eastern data. However, in general, the data bases collecting pollution measurements are often statistically poor: only a limited amount of data is available to estimate the relevant parameters of interest here and the data themselves need to be checked with great accuracy to avoid misuse of information.

Then, we outlined the mathematics of fractals and multifractals, presenting both the theoretical framework and the practical analysis techniques; these are the used to investigate the available data. The geometrical monofractal analysis of the sampling networks (both for radioactivity and rainfall) in four Countries (i.e., Austria, Czechoslovakia, Northern Italy and Western Germany) shows their fractal nature over a relevant range of scales; such a study confirms the sparseness and the inhomogeneity of the monitoring structure, and provides a measure of the amount of information (almost surely) "lost" in the gaps of such networks.

The study of the data statistics in the same four Countries shows the presence of a Self Organised Criticality (hyperbolic) behaviour; as a consequence, strong fluctuations (i.e. the "hot spots" found in the available data) are expected even at a theoretical level. Also, the problem of the divergence of moments may affect the analysis, indicating the necessity of a multifractal characterisation of the phenomenon. In fact, the results strongly support the multifractal nature of the phenomena investigated; even more, the underlying multifractal dynamics may be classified as an "unconditionally hard" multifractal process. The same considerations apply also in the case of the radioactive concentration data collected in the region surrounding the Chernobyl nuclear plant.

Overall, using universal multifractals we achieved two main results. On the one hand, the statistical characterisation of the phenomenon is obtained simply by means of easily-implementable procedures. On the other hand, we gain a deeper insight into the intermittent behaviour of the data distribution at all scales: the wild fluctuations are not regarded as anomalous and discarded, on the contrary they are kept as an essential feature of the phenomenon. The former conclusion has mainly implications for generating fast computer codes (e.g., in case of environmental pollution accidents due to radioactive release). Moreover, there might be further important theoretical and practical consequences in environmental sciences: in fact, we provided a relatively simple way to study the statistics of sparse data, preserving their intrinsic features such as intermittency (i.e. no "a priori" regularity or smoothness hypotheses are required as in conventional objective analyses), and we investigated the phenomenon at all scales, from the strictly local concentration up to the largest spatial average.

Finally, we feel important to stress that the estimates of the multifractal parameters allows to generate stochastic simulations. For the purpose of the analysis, as an offspring of the present Contract, a special software (named "Multifractalia") performing the multifractal analysis of 1D and 2D data has been provided and it is freely available together with the Final Report. Clearly, exploiting such a parametrization and the mathematics of multifractals it might be possible in the near future to create statistical procedures able to estimate the intensity of a field (e.g. radioactive pollution) even in the "gaps" of a (multi) fractal network, possibly exploiting the knowledge of other fields (such as rainfall, wind, ...). the original techniques outlined represent a first contribution to the ongoing researches concerning the estimate of multifractal fields and may have wide and fruitful applications in many areas of environmental sciences and risk assessment, since natural phenomena are often sampled over sparse networks.

In conclusion, we believe that the work done in this Project not only demonstrates the relevance of multifractal notions and techniques to assess and monitor radioactive fallout, but also supports the feasibility of the corresponding routine techniques. By means of the free software provided together with the Final Report, in spite of the underlined difficulties, one may already proceed to analyse and forecast experimental data as outlined above.

MULTIFRACTAL ANALYSIS AND SIMULATION OF CHERNOBYL FALL-OUT IN EUROPE

SUMMARY

THE RESEARCH INITIATED WITHIN THE STUDY GROUP "POST CHERNOBYL ACTION 1 : IMPROVEMENTS OF RELIABLE LONG DISTANCE ATMOSPHERIC TRANSPORT MODELS (PAVIA C.E.A. FONTENAY AUX ROSES, R.D.A. WARRINGTON, DEMOKRITOS ATTIKI : PAVIA B16*-0241-I(A)) AND CONTINUED WITH THE STUDY CONTRACT "EMPIRICAL AND FRACTAL DESCRIPTION OF RADIOLOGICAL DATA DETECTED AFTER THE CHERNOBYL ACCIDENT" (PRAVDA B17*-CT90-0062).

TOWARDS THE END OF THE FIRST CONTRACT WE PROPOSED A DESCRIPTIVE MONOFRACTAL MODEL AS A FIRST APPROXIMATION TO THE DESCRIPTION OF THE AIR RADIOACTIVITY DISTRIBUTION IN NORTHERN ITALY, WHERE CERTIFIED MEASUREMENTS HAD BEEN COLLECTED. THE RIGIDITY OF THE BASIC ASSUMPTIONS BECAME SOON CLEAR AND THE NECESSITY OF A MORE FLEXIBLE APPROACH BECAME EVIDENT.

IN THE SECOND CONTRACT (IN PROGRESS : END OCTOBER 1992) WE WILL PROVIDE A MULTIFRACTAL ANALYSIS OF THE CUMULATIVE DEPOSITION OF 137 CS IN NORTHERN ITALY AND SOME EUROPEAN COUNTRIES IN ORDER TO CALCULATE THE CHARACTERISTIC PARAMETERS TO BE USED IN FURTHER MODELING, BASED ON THE ORIGINAL IDEA OF UNIVERSAL MULTIFRACTALS PUT FORWARD AND S.LOVEJOY IN RECENT YEARS. BASED ON THE PRELIMINARY RESULTS NEARLY TO BE OBTAINED AND THE SOUNDNESS OF THE APPROACH, WE ARE CONFIDENT THAT WITHIN THE TERMS OF A NEW TWO-YEAR CONTRACT IT SHOULD BE DEFINITELY POSSIBLE TO FORMULATE A PRACTICAL MULTIFRACTAL MODEL ABLE TO PROVIDE A REASONABLE DESCRIPTION OF AIR POLLUTION AND OR CUMULATIVE DEPOSITION IN SOME EUROPEAN COUNTRIES. TO THIS PURPOSE, THE COLLABORATION IS EXTENDED TO INCLUDE METEO FRANCE; THE EXTERNAL COLLABORATION OF THE GROUP LEAD BY PROF. SHAUN LOVEJOY (MC GILL UNIVERSITY, MONTREAL IS FORESEEN.

FOR FURTHER DETAILS SEE ATTACHED TECHNICAL ANNEX.