Concentration Modeling: Generalized Cantor Sets

Mark E. Gettings
U.S. Geological Survey

Phillip Hammonds
Northrop-Grumman Information Technology

Bernard P. Zeigler
Arizona Center for Integrative Modeling & Simulation

In one dimension, the multiplicative cascade can be viewed as a conservative system in which the contents are distributed into ever-decreasing size line segments by putting a portion of the contents of a line segment into each of the segments.  In the limit the set becomes one of an infinite number of members of infinitesimal size, each of infinite density. If the generation of the set is stopped at a finite number of steps (finite scale), the process models concentration of the contents on a line. The process can be generalized into unequal length segments and unequal (and nonzero) proportions allocated to each segment. This defines a general multiplicative process in which the final content of each segment is a product of the proportions from all the steps or generations of line division in the final distribution. If the order of applying the division and proportioning is randomized, distributions statistically identical to real distributions observed in earth science can be obtained. Two examples of application of the concentration model are the distribution of tonnage of ore as a function of deposit grade and the distribution of magnetic susceptibility in hydrothermally altered rocks in porphyry systems. In the latter model, the line division process determines the distribution of fractures. The susceptibility of interfracture blocks is modeled by oxidation of the wallrock, a process controlled by the diffusion equation. Multiple processes may be modeled by serial application of line division processes. The model can be further generalized to non-conservative systems if the sum of the proportions distributed to segments is not unity.  Sums greater than one represent systems accumulating mass from outside the system, while sums less than one represent systems losing mass. The concentration of selenium in the San Francisco Bay ecosystem can be modeled by such a series of superposed (non-conservative) line division models. The resulting distributions are complex and resist inversion to yield the concentration process parameters.

Created by Mathematica  (April 20, 2004)