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.

Disaster-originated placename is a kind of disaster subculture that is used for a practical purpose of identifying a location while reminding the past disaster experience. They are expected to transmit the risks and knowledge of high-risk low-frequency natural hazards, surviving over time and generations. This paper compares the perceptions to tsunami-originated placenames in local communities having realistic and exaggerated origins in Sanriku Coast, Japan. The reality of tsunami-originated placenames is first assessed by comparing the tsunami run-ups indicated in the origins and that of the tsunami in the Great East Japan Earthquake 2011 using GIS and digital elevation model. Considerable proportions of placenames had exaggerated origins, but the group interviews to local communities revealed that origins indicating unrealistic tsunami run-ups were more believed than that of the more realistic ones. We discuss that accurate hazard information will be discredited if it contradicts to the people’s everyday life and the desire for safety, and even imprecise and ambiguous information can survive if it is embedded to a system of local knowledge that consistently explains the various facts in a local area that requires explanation.

Fractal dimension (FD) has been widely used to measure the spatial complexity of human settlements, primarily through 2D analyses of urban form. While 2D FD captures space-filling properties consistent with hierarchical organization such as that in central place theory, it does not take into account the highly uneven and multi-scalar nature of population density. A 3D FD incorporating population counts offers additional structural information. Therefore, this study investigated how 3D FD relates to settlement structure. First, we analyzed theoretical settlement configurations—including regular, clustered, and hierarchical patterns—that can be systematically morphed using structural parameters. Hierarchical systems are constructed following central place theory and population distribution consistent with the rank–size rule, and they are controlled by parameters such as the branching factor (K-value) and degree of metropolitan dominance. We estimated 3D FD for both these configurations to demonstrate how FD varies systematically with changes in hierarchical organization. Results showed that hierarchical patterns consistently yielded FD values greater than 2, while regular or non-hierarchical patterns approached or remained at 2. While FD cannot be defined in the most sparsely populated areas, rural areas exhibited FD greater than 2 as well as the highest FD. Second, building upon these insights, we estimated local 3D FD from gridded population data for the United States using a moving-window approach (400 km × 400 km). The resulting maps revealed substantial regional variation in settlement structure. Large metropolitan regions associated with polycentricity exhibited smaller FD values than smaller cities, while the FD of rural areas varied considerably. Additionally, we measured the fractal depth and the range of scales at which fractal properties are present. Rural areas generally have thinner fractal depth, although those in the arid West were even thinner. These findings provide a structurally grounded interpretation of 3D FD and demonstrate its utility for analyzing settlement systems relevant to the whole urban−rural continuum.