Multifractality and spatial statistics

The concepts of fractals and multifractals have been increasingly applied i n various fields of science for describing complexity and self-similarity i n nature. Fractals and multifractals are a natural consequence of self-simi larity resulting from scale-independent processes. In the present paper, a theoretical investigation is developed to illustrate: (1) the characteristi cs of multifractality as measured by the parameter tau "(q); (2) relationsh ips between multifractality and spatial statistics including semivariogram and autocorrelation in geostatistics, indexes used in lacunarity analysis a nd correlation coefficients. It can be shown that these statistics primaril y are related to multifractality as determined by tau "(1). This is an impo rtant result because not only does it provide the link between multifractal s and spatial statistics but it also shows that statistics based on second- order moments are restrictive in that they only characterize a multifractal measure around the mean value. In applications where extreme values need t o be taken into account, the entire multifractal spectrum should be used ra ther than local properties of the spectrum around the mean only; alternativ ely, statistics defined on the basis of higher-order moments can be employe d for analysis of extreme values. These theoretical results are illustrated by means of application to Landsat TM imagery (bands 1 to 7) from the Mitc hell-Sulphurets mineral district, northwestern British Columbia, Canada. It is shown that variations of the TM data bands 1 to 3 in this area can be a pproximated by fractals but for those of bands 4, 5 and 7, multifractal mod els with different fractal spectra must be used.:The multifractality of the se images is evaluated and several methods of spatial analysis are applied to the dataset including a new version of principal component analysis in c onjunction with the newly defined statistical parameters based on multifrac tality. (C) 1999 Elsevier Science Ltd. All rights reserved.