Unified Metrics for Characterizing the Fractal Nature of Geographic Features

The term fractal is used to describe an object displaying self-similarity at different scales. This self-similarity can be measured by either the power-law exponent or the ht-index, which is a recently proposed method for characterizing the fractal nature of geographic features. Although increasingly popular in geography, the ht-index is not sensitive to changes or evolutions of fractals, limiting its usefulness as an alternative “fractal dimension” to the power-law exponent. Two improvements to the ht-index were suggested in the literature, namely, the cumulative rate of growth (the CRG index) and the ratio of areas in a rank-size plot (the RA index). The CRG index is sensitive but not monotonic, however, with respect to the evolution of fractals. The RA index is both sensitive and monotonic but not interpretable in fractal terms. In this article, two novel metrics, referred to as unified metrics, are proposed by combining advantages of the ht-index and all of its improvements, being simultaneously easy to interpret, monotonic with respect to the evolution of fractals, and sensitive to changes in the evolution. The usefulness of unified metrics was demonstrated by both numerical experiments and case studies. Given that the idea behind the ht-index has led to a relaxed, emerging popular definition of fractals, the proposed unified metrics have great potential to be used as the standard fractal dimension along with such a definition.