Statistics of 2D Cuts through 3D Fractal Objects, with Application to Xenoliths in Plutons
The origin of xenoliths in plutons is of great interest in structural geology and petrology. If xenoliths form by fragmentation (e.g., by thermal shock), they should have a fractal size distribution with D ~2.5 (Turcotte, JGR, 1986). Conversely, if they form by dicing of wall rock by dikes, they should have a non-fractal size distribution (Glazner and Bartley, GSAB, 2006). Processes such as melting and disaggregation that occur after xenolith incorporation can modify the initial distribution and complicate the picture.
Measuring the size distribution of xenoliths is complicated by collection of data on 2-dimensional cuts (outcrop surfaces). Mandelbrot conjectured that such a cut will reduce the fractal dimension by 1, and numerical experiments confirm this. We generated spheres with fractally distributed diameters with D3 = 2.5, distributed them uniformly in the vertical dimension, and then cut the resulting array with a horizontal plane. Diameters of the resulting circles have a fractal dimension D2 ~1.5. The physical basis for this is straightforward; smaller particles are significantly less likely to intersect an arbitrary plane than larger ones. We found D2 ~1.5 for a limited xenolith dataset from Yosemite National Park. In contrast, Farris and Paterson (Can Min, 2007) argued that examining 2D cuts will have little effect on the derived fractal dimension, and found D2 ~2.0-2.5 for several samples, implying that D3 would be ~3.0-3.5. This large value of D2 probably stems from their measurement method (image thresholding) which can produce a large number of spurious small "xenoliths."
These results have significant implications for outcrop examination. For D3 = 2.5, there should be ~100,000 1 cm xenoliths for every 1 m xenolith. However, in 2 dimensions, the observed ratio will only be ~1000. Thus, the apparent scarcity of small xenoliths may result from fractal statistics.