Self-similarity and multiplicative cascade models
F Agterberg
In his 1978 monograph ‘Lognormal-de Wijsian Geostatistics for Ore
Evaluation’, Professor Danie Krige emphasized the scale-independence
of gold and uranium determinations in the Witwatersrand goldfields. It
was later established in nonlinear process theory that the original model
of de Wijs used by Krige was the earliest example of a multifractal
generated by a multiplicative cascade process. Its end product is an
assemblage of chemical element concentration values for blocks that are
both lognormally distributed and spatially correlated. Variants of de
Wijsian geostatistics had already been used by Professor Georges
Matheron to explain Krige’s original formula for the relationship
between the block variances as well as permanence of frequency distributions
for element concentration in blocks of different sizes. Further
extensions of this basic approach are concerned with modelling the
three-parameter lognormal distribution, the ‘sampling error’, as well as
the ‘nugget effect’ and ‘range’ in variogram modelling. This paper is
mainly a review of recent multifractal theory, which throws new light on
the original findings by Professor Krige on self-similarity of gold and
uranium patterns at different scales for blocks of ore by (a) generalizing
the original model of de Wijs to account for random cuts; (b) using an
accelerated dispersion model to explain the appearance of a third
parameter in the lognormal distribution of Witwatersrand gold determinations;
and (c) considering that Krige’s sampling error is caused by
shape differences between single ore sections and reef areas.
Keywords: lognormal-de Wijsian geostatistics, self-similarity, multifractals, nugget
effect, Witwatersrand goldfields.