Box Height-Independent Differential Bar Cumulation (DBC) for 3D Raster Surface Fractal Dimension Analysis

Abstract The box-counting method is the predominant technique for calculating fractal dimensions (FD) for raster data. However, past studies focused primarily on measuring the FD of 2D binary raster. While fields like image analysis have advanced with methods such as differential box counting (DBC) to measure the FD of 3D surfaces, this approach has yet to be applied to geographical raster data, mainly due to challenges like various data depth formats, choice of box height, and type of distribution of raster values within data. In this study, we address these challenges, particularly focusing on selecting an appropriate box height which is also a methodological issue even for images. The difficulties in DBC, such as too-small and too-large box height problems, appear on finer and coarser scales of log–log plots, respectively, and there are optimal scales that represent the accurate FD somewhere in the middle. We proposed key modifications to the conventional DBC method theoretically to reduce the number of inappropriate scales due to the difficulties in DBC. To identify the effective scales in log–log plot, we introduced a novel visualization technique which can be used to detect subtle variations in log–log plots by visual inspection, aiding in the identification of box height problems and related difficulties. We applied and compared both the conventional and modified DBC method to two empirical datasets, population data and digital elevation model data, across the USA. Our results demonstrate that the log–log plots display greater consistency, and the FD values converge to a larger value as the box height decreases. The enhanced log–log plots from the results showed that there is no too-small box height problem at all, and the box height is smaller the better. Based on the empirical results, we proposed a derived method that eliminates the need for selecting a specific box height, addressing all three difficulties of DBC identified in this study. Since the derived method no longer uses the ceiling function and does not count boxes, the acronym DBC should now stand for differential bar cumulation.