Multifractals and geostatistics

Abstract

In several recent studies, 2-dimensional applications of local singularity analysis including regional studies based on stream sediment data show local minima that are spatially correlated with known mineral deposits. These minimal singularities, which may provide targets for further mineral exploration, generally are smoothed out when traditional geostatistical contouring methods are used. Multifractal analysis based on the assumption of self-similarity predicts strong local continuity of element concentration values that cannot be readily determined by variogram or correlogram analysis. This paper is concerned with multifractal and geostatistical modeling of the largest and smallest geochemical element concentration values in rocks and orebodies. These extreme values correspond to local singularities with near-zero fractal dimensions that occur close to the minimum and maximum singularity in the multifractal spectrum. The latter cannot be determined by means of the method of moments because of small-sample size problems arising when the largest and smallest concentration values are raised to very large powers q. It is shown by means of a computer simulation experiment and application to copper determinations from along the 7-km deep KTB borehole in southeastern Germany, that local singularity analysis can be used to determine all singularities including the extreme values. The singularities estimated by this method are linearly related to logarithmically transformed element concentration values. This simple relation also can used to measure the small-scale nugget effect, which may be related to measurement error and microscopic randomness associated with ore grain boundaries.