A design‐based proof of Wicksell's integral equation

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Abstract
SUMMARY: Wicksell's integral equation (1925) describes, for an aggregate of spherical particles, the relation between the distribution of sphere diameters and the distribution of diameters of circular profiles observed on a planar section. Usually, this integral equation is proved under the assumption that the particle aggregate is a realization of a stationary stochastic process while the position of the planar probe is allowed to be arbitrary. In this note, it is shown that, under the alternative assumption that diameters of circular profiles are sampled by means of a planar FUR (fixed orientation uniform random) probe, Wicksell's equation holds for a deterministic aggregate of spherical particles with arbitrary positions and sizes. Since the proof is based on the randomness generated by the sampling (the design), the proof may be characterized as design‐based. It can thus be concluded that Wicksell's equation may be applied to any aggregate of spherical particles if diameters are sampled by means of random (FUR) planar probes.

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