Letter to the Editor

Mathematical Geosciences -     

Geostatistical case studies of the advantages of lognormal-de Wijsian kriging with mean for a base metal mine and a gold mine

Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.     

The gravitational field of a static infinite sheet of matter

The solution of the Einstein field equations in the case of an infinite, static, uniform distribution of matter lying parallel to thex–y plane is obtained. This metric is equivalent to the uniformly accelerated metric which causes the particles to move with uniform acceleration parallel to thez-axis.     

Wave and stability properties of black holes

In this paper we transform the wave equation governing gravitational perturbations of a Schwarzschild black hole from its standard Schrödinger or Regge-Wheeler form to a Klein-Gordon type wave equation. This latter form reveals immediately that incoming waves with frequencies () _cml , a critical frequency, are completely reflected (transmitted). This process is entirely due to the radial variation of the cut-off frequency inherent in the dispersive nature of the wave propagation properties of gravitational perturbations of the Schwarzschild metric. Moreover, those high-frequency waves ( _cml) which penetrate through the region near the Schwarzschild radiusr _sare, on crossing this event horizon, attenuated by a factor exp (–r _s/c), thereby dumping most of their energy and momentum into the black hole. It is shown that in the vicinity ofr _sthe metric is locally unstable. This feature and the wave absorption process indicate that the neighbourhood aroundr _sis dynamically active, and, as well as acting like a Hawking-type particle creator, will behave as a wave emitter in order to relax the stresses on the metric.     

Studies of the effects of outliers and data transformation on variogram estimates for a base metal and a gold ore body

Variograms for gold and lead values from the Loraine and Prieska mines, respectively, indicate that data outliers can seriously distort and/or mask the real variogram patterns. Studies show that this problem is best overcome for these mines by logarithmic transformation of the data, and/or a suitable screening out of such outliers, and/or more robust variogram estimation procedures; the benefits are particularly significant when the basic data is limited.