Fractal descriptions for spatial statistics

Measures of spatial statistics have been available for estimating means, calculating or assessing differences, estimating nearest neighbor distances, and such, but have not provided a general approach to describing variances. Because measures of heterogeneity depend upon choosing a particular element size in the domain, estimates of apparent heterogeneity are larger with high-resolution observations than with low-resolution data. Many descriptors might be used to describe the relationships between apparent heterogeneity and the size of the observed spatial elements, but we have found that fractal relationships provide concise and precise descriptions of many types of data over large ranges of element sizes. Perhaps more importantly, the fractal approaches give additional insight, such as measures of spatial correlation, and often suggest ways of approaching the underlying basis of the heterogeneity.