The annual earthquake predictions of the China Seismological Bureau (CSB) are evaluated by means of an R score (an R score is approximately 0 for completely random guesses, and approximately 1 for completely successful predictions). The average R score of the annual predictions in China in the period 1990–1998 is about 0.184, significantly larger than 0.0. However, background seismicity is higher in seismically active regions. If a ‘random guess' prediction is chosen to be proportional to the background seismicity, the expected R score is 0.123, and the nine-year mean R score of 0.184 as observed is only marginally higher than this background value. Monte Carlo tests indicate that the probability of attaining an R score of actual prediction by background seismicity based on random guess is about . It is concluded that earthquake prediction in China is still in a very preliminary stage, barely above a pure chance level.

We discuss necessary and sufficient conditions for power-law and polynomial models to be correlation functions on bounded domains. These results date back to unpublished work by Matheron (1974) and generalize the findings of Gneiting (1999).

A popular procedure in spatial data analysis is to fit a line segment of the form c(x) = 1 - α ||x||, ||x|| < 1, to observed correlations at (appropriately scaled) spatial lag x in d-dimensional space. We show that such an approach is permissible if and only if

the upper bound depending on the spatial dimension d. The proof relies on Matheron's turning bands operator and an extension theorem for positive definite functions due to Rudin. Side results and examples include a general discussion of isotropic correlation functions defined on d-dimensional balls.

We formulate stochastic indicator parameters that characterize pollution levels in geographical regions with heterogeneous contaminant distributions. The indicator parameters are expressed in terms of the random fields representing the contaminant distributions and the critical threshold level specified by health and environmental standards. Certain theoretical results are proven regarding univariate and bivariate indicator parameters. The analytical expressions obtained are general and can be used in practice for various types of contaminant distributions. A test of ergodicity-breaking is suggested for scientific and engineering applications in terms of the indicator parameters. Fractal characteristics of the indicator parameters are discussed. The effects of modelling and observation scale on exceedance contamination analysis are examined. Indicator random field parameters are studied on both continuum and lattice domains using analytical means and numerical simulations.