More on the Corpuscle Problem

Wicksell was a statistician that was called in to work on an interesting problem in which a researcher was attempting to characterize what appeared to be spherical objects in the tissue.

One of the first tasks Wicksell did was to use exhaustive serial sections to verify that the objects were indeed spherical. He took diameter measurements from sections and determine that he was dealing with a population of spherical objects.

There were things that he was not able to determine, but appeared to be valid. For instance, the distribution of the spheres appeared to be uniformly random. Another issue was that the object sizes were not correlated.

Examples of such problems in geology would be seen in conglomerates. These are rocks with pebbles. If the water moving pebbles slowed down over time then the pebbles would change from larger pebbles at the bottom to smaller pebbles at the top. The distribution would not meet the uniformly random requirement. If there were a few larger pebbles and they were surrounded by smaller pebbles caught in eddies, then there would be a correlation between large pebbles and smaller pebbles that would make the math derived by Wicksell invalid.

There are a number of mathematical assumptions that are well stated in Wicksell’s 1925 paper.

Wicksell goes on to provide a solution that only works with spherical objects under a list of important assumptions that were likely to be true for the tissue the researcher was studying.

This made it possible to not only determine the number of spheres, but just as important it provided a size distribution for the spheres. In other words, it showed what fraction was small, and what fraction was large.

Does Wicksell’s work address shape? Very clearly it only addresses a population of spheres.

Does Wicksell’s work address the long simmering issue for geologists on section orientation? No. He is only dealing with spheres.

Does Wicksell’s work address orientation? No since spheres cannot be oriented. In fact, Wicksell goes on to show that there is no solution for triaxial ellipsoids.

Does Wicksell’s work address size? Obviously, since he provides a size distribution.

Wicksell knew that profiles are not numbers of objects. He did not count profiles or profiles per unit area. He was interested in the observed profile diameters. That’s very different. The distribution of observed profile diameters is related to the size distribution of the original spherical population. Through numerical unfolding the original size distribution can be determined and that leads to a determination of the population size.

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