Computing fractal dimension at large geographic scales using Discrete Global Grid Systems

Various empirical methods exist to calculate fractal dimension of geospatial objects with the box-counting principle being a popular one. However, these methods generally require geospatial data to be projected to Euclidean space. While this works fine at small geographic scales, computation at larger or global scales introduces distortions inevitable with projection due to the curvature of the earth. I show from mathematical principles how Discrete Global Grid Systems (DGGSs) – hierarchical spatial data structures composed of polygonal cells that are increasingly being used for modelling geospatial data – can be employed creatively to act as the covering set for calculating the Minkowski-Bouligand dimension using the box-counting principle. This enables computation of the fractal dimension of geospatial data in spherical coordinates without having to project the data in question on a planar surface. Results on synthetic datasets are within 1% of their theoretical fractal dimensions. A case study on opaque cloud fields obtained from a geostationary meteorological remote sensing satellite image yields a result of 1.577±0.0207 when aggregated using three different geodesic DGGSs based on the Icosahedral Snyder Equal Area (ISEA) projection, in line with values reported in the literature. As the cells of a DGGS are generally pre-defined and fixed to the earth, this method also brings some relief associated with the box-counting method in general, particularly the choice of cell-sizes to be sampled as well as the placement and orientation of the grid that acts as the covering set – issues that are usually circumvented by rules of thumb and conventions. I comment on the possibility to extend the method for use with raster data.  Ways to improve the method using low-aperture DGGSs to better capture the self-similarity and possibilities of developing custom DGGSs for this purpose are also noted. Being a computationally intensive method, development of software libraries making use of parallel computing to enhance performance and scalability is also proposed. With climatic variables exhibiting spatiotemporal autocorrelation with long-range effects, I believe this method would be of interest to climate scientists interested in studying their fractal properties at continental and global scales.