Abstract
We consider a stationary and isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with value 0 or 1 according to whether a regular stationary process X is less or greater than a given level. The intervals corresponding to the points at which X is in a given phase are named chords. We are interested in obtaining information on the chord-length distribution functions. Working with the Palm probability measure and using level crossings techniques, in particular Rice methods, we can obtain not only the exact analytical formula of the chord-length distribution function but also the joint distribution function of the lengths of two successive chords. Finally, we indicate some concrete applications for the computation of usual stereological parameters.
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This work was partially supported by the French grant “Mipomodim” ANR-05-BLAN-0017.
Marie Kratz is also a member of MAP5, Univ. Paris Descartes.
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Estrade, A., Iribarren, I. & Kratz, M. Chord-length distribution functions and Rice formulae. Application to random media. Extremes 15, 333–352 (2012). https://doi.org/10.1007/s10687-011-0141-y
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DOI: https://doi.org/10.1007/s10687-011-0141-y