Eur. Phys. J. B 47, 397-409 (2005)
https://doi.org/10.1140/epjb/e2005-00338-5

Second-order analysis by variograms for curvature measures of two-phase structures

¹ Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia
² Institut für Stochastik, Friedrich-Schiller-Universität Jena, 07740 Jena, Germany
³ Institut für Theoretische Physik Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany
⁴ Institut für Stochastik, TU Bergakademie Freiberg, 09596 Freiberg, Germany

Corresponding author: ^a stoyan@orion.hrz.tu-freiberg.de

Received: 29 March 2005
Published online: 28 October 2005

Abstract

Second-order characteristics are important in the description of various geometrical structures occurring in foams, porous media, complex fluids, and phase separation processes. The classical second order characteristics are pair correlation functions, which are well-known in the context of point fields and mass distributions. This paper studies systematically these and further characteristics from a unified standpoint, based on four so-called curvature measures, volume, surface area, integral of mean curvature and Euler characteristic. Their statistical estimation is straightforward only in the case of the volume measure, for which the pair correlation function is traditionally called the two-point correlation function. For the other three measures a statistical method is described which yields smoothed surrogates for pair correlation functions, namely variograms. Variograms lead to an enhanced understanding of the variability of the geometry of two-phase structures and can help in finding suitable models. The use of the statistical method is demonstrated for simulated samples related to Poisson-Voronoi tessellations, for experimental 3D images of Fontainebleau sandstone and for two samples of industrial foams.