ARE SMALL PLATES FRACTAL? Message on Peter Bird's office door (Geology 4656) The plates size (in steradians) distribution has been studied by Bird (2003, Fig. 19) and by Sornette and Pisarenko (2003), using Bird's preliminary results. Bird found that the distribution for 52 plates has two branches: a scale-invariant for small plates with a power-law index D=0.33 and a rapidly decaying tail for large plates. The results obtained by Sornette and Pisarenko are similar with fractal exponent value 0.25. Clearly, it would be more difficult to unambiguously identify even smaller plates -- their statistics would not be complete and, finally, the 3-D deformation pattern would need to be taken into account for plates whose size is comparable with the thickness of a seismogenic layer. Thus, how should we proceed to answer the question above? Another piece of evidence that supports the fractal character of Earth deformation comes from spatial distribution of shallow earthquakes. Kagan and Knopoff (1980) investigated the distribution of distances between earthquake pairs for several catalogs. For global catalogs the appropriately scaled distribution has again two branches -- scale-invariant for small distances, arguably from zero up to 2000-3000 km, and a non-fractal long-distance tail (2000-20,000 km). The fractal (correlation) dimension (d) is for 3-D about 2.2 (also see Kagan, 1991). An update of the global figure (Fig. 8 in Kagan, 2005) is posted http://moho.ess.ucla.edu/~kagan/p9_p2rev.pdf. It would be interesting to match these two distributions, to see which fractal dimension exponent corresponds to the tectonic plates distribution. The only method that could be effectively applied to solve this problem is most likely simulation. Plates formation could be represented as Voronoi tessellation of a sphere. Many theoretical and computer studies exist for a Voronoi tessellation of a sphere based on the Poisson point distribution (Okabe et al., 2000). However, I am not aware whether anyone has tried to model a tessellation based on a fractal point pattern on a sphere. Kagan and Knopoff (1980, Figure 11) obtained the correlation dimension estimate for tessellation of sphere by three regular cubic (i.e., with triple junctions) polyhedra. As one can easily guess, the correlation dimension for small distances (0-3000 km) is 1.0 (or 2.0, if one takes depth into account, a transition to the large-range pattern occurs at about 2000-5000 km, depending on the polyhedron used. The small-range dimension value corresponds to long edges of regular polyhedra: close pairs of points are more likely to belong to the same cell edge. For regular polyhedra tessellation the plates distribution displayed similarly to Bird's (2003) Fig. 19 and to Sornette and Pisarenko' (2003) diagram, would have a step-function at \pi, 2\pi/3, and \pi/3 for tetrahedron, cube, and dodecahedron, respectively. The left-hand distribution exponent (D) is zero. Clearly, if we introduce a fractal distribution of plate sizes, the correlation dimension should increase, since in addition to long edges of large plates, the numbers of short-range point pairs would increase due to the branching edges of smaller plates. Would this be sufficient to increase the d-dimension value from 1.0 to 1.25 (or from 2.0 to 2.25 in the 3-D case)? Or is d=D+2, in general? Even if the above equality is true, the estimates of the correlation dimension are dependent on the time span of a catalog, but for a sufficiently long catalog the d-exponent should approach an asymptotic value (Kagan, 1991). Whereas the fractal exponent for plate size distribution is a result of million of years of plate tectonics evolution, the d-value is determined for catalogs covering a few decades, therefore d=2.2 is most likely to be an under-estimate. On the other hand, plates are not rigid, most of them have significant intraplate seismic activity, which should cause an increase of the d-value. Several simulation tests of increasing difficulty can be envisioned. 1. Fractal Voronoi tessellation of a plane -- calculating the connection between the fractal dimension of cell sizes and the correlation dimension of the random points on the cell edges. It is not yet clear how to simulate a fractal ensemble of points -- Levy flight is one possibility, critical percolation (Hinrichsen and Schliecker, 1998) is another. 2. Same for a sphere. 3. Similar to item 2, but starting a fractal pattern simulation after major plates have been selected using a Poisson algorithm. This should reproduce two plate populations -- large plates and small fractal ones. 4. Assigning Euler poles to each of the plates so that plate tectonics on a sphere could be simulated. Depending on the type of deformation for each plate edge, the density of points (epicenters) can be adjusted according to Table 2 in Bird and Kagan (2004). This would represent a kinematic solution for the problem. 5. Finally, a dynamic solution can be contemplated -- simulating mantle convection and, depending on an assumed interaction between the mantle and the crust, obtaining a distribution of tectonic plate sizes. In addition to the fractal exponent (d), the transition from a fractal pattern to a large-range pattern should be explored -- in the epicenter/hypocenter distribution plot (Fig. 8 in Kagan, 2005). These are the vertical and horizontal (distance) coordinates of the change-over point. REFERENCES: Bird, P., 2003. An updated digital model of plate boundaries, Geochemistry, Geophysics, Geosystems, 4(3), 1027, doi:10.1029/2001GC000252. http://element.ess.ucla.edu/publications/2003_PB2002/2003_PB2002.htm Bird, P., and Y. Y. Kagan, 2004. Plate-Tectonic Analysis of Shallow Seismicity: Apparent Boundary Width, Beta, Corner Magnitude, Coupled Lithosphere Thickness, and Coupling in Seven Tectonic Settings, Bull. Seismol. Soc. Amer., 94(6), 2380-2399. Hinrichsen H, Schliecker G., 1998. Scaling laws and topological exponents in Voronoi tessellations of intermittent point distributions JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 31 (24): L451-L455. Kagan, Y. Y., 1991. Fractal dimension of brittle fracture, J. Nonlinear Sci., 1, 1-16. Kagan, Y. Y., 2005. Earthquake spatial distribution: Correlation dimension, work in progress, see http://scec.ess.ucla.edu/~ykagan/p2rev_index.html Kagan, Y. Y., and Knopoff, L., 1980. Spatial distribution of earthquakes: The two-point correlation function, Geophys. J. R. astr. Soc., 62, 303-320. Okabe, A., Boots, B., Sugihara, K., and Chiu, S., 2000. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed., Wiley, Chichester, 671 pp. Sornette, D., and V. Pisarenko, 2003. Fractal plate tectonics, Geophys. Res. Lett., 30(3), 1105, doi:10.1029/2002GL015043