Statistical properties of ecological and geologic fractals

Abstract

To use fractal models for ecological and geologic data, the statistical properties of fractals need to be clarified. No sampling or estimation theory for fractals currently exists. Several concrete steps in this direction are taken here. First, the information fractal dimension is proposed as a new measure that is relatively robust with respect to sampling error and can handle intensive data. The information fractal is tested with field data and is shown to be capable of delineating stratified structures and defining the scale of heterogeneity in the data. Comparison to semivariance analysis reveals the superiority of the fractal model for sample data that are nonisotropic and nonstationary. It is argued that approaches using regression to estimate fractal dimensions of spatial patterns are statistically invalid, and alternatives are proposed. Sampling of natural objects with transects (e.g., wells) is explored. For nonisotropic media (or maps), random placement of transects is shown to give an unreliable estimate of pattern. For transects taken perpendicular to a directional pattern (i.e., strata), it is shown that the mean of multiple estimates of the multiscale fractal dimensional profile does converge to the true value. Other sampling issues are addressed.