Ht-Index for Quantifying the Fractal or Scaling Structure of Geographic Features

TLDR

Geographic features such as mountains and coastlines are fractal, yet some studies question this universality because the strict definition of fractal dimension may not capture complexity. In this article, we propose an alternative, ht‑index, to quantify the fractal or scaling structure of geographic features. The ht‑index is defined by the recurrence of far‑more small than large elements across scales, and the authors illustrate its application through three case studies and discuss its complementarity to fractal dimension. The higher the ht‑index, the more complex the geographic feature.

Abstract

Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is dubious, is mainly attributed to the strict definition of fractal dimension as a measure or index for characterizing the complexity of fractals. In this article, we propose an alternative, ht-index, to quantify the fractal or scaling structure of geographic features. A geographic feature has ht-index (h) if the pattern of far more small things than large ones recurs (h – 1) times at different scales. The higher the ht-index, the more complex the geographic feature. We conduct three case studies to illustrate how the computed ht-indexes capture the complexity of different geographic features. We further discuss how ht-index is complementary to fractal dimension and elaborate on a dynamic view behind ht-index that enables better understanding of geographic forms and processes.