Abstract
Discrete Global Grid Systems (DGGSs) are spatial references that use a hierarchical tessellation of cells to partition and address the entire globe. They provide an organizational structure that permits fast integration between multiple sources of large and variable geospatial data sufficient for visualization and analysis. Despite a significant body of research supporting hexagonal DGGSs as the superior choice, the application thereof has been hindered owing in part to the lack of a rational hierarchy with an efficient addressing system. This paper presents an algebraic model of encoding scheme for the Aperture 3 Hexagonal (A3H) DGGS. Firstly, the definition of a grid cell, which is composed of vertices, edges, and a center, is introduced to describe fundamental elements of grids. Secondly, by identifying the grid cell with its center, this paper proves that cell centers at different levels can be represented exactly using a mixed positional number system in the complex plane through the recursive geometric relationship between two successive levels, which reveals that grid cells are essentially special complex radix numbers. Thirdly, it is shown that through the recursive geometric relationship of successive odd or even levels, the mixed positional number system can also be applied to uniquely represent cell centers at different levels under specific constraint conditions, according to which the encoding scheme is designed. Finally, it is shown that by extending the scheme to 20 triangular faces of the regular icosahedron, multi-resolution grids on closed surfaces of the icosahedron are addressed perfectly. Contrast experiments show that the proposed encoding scheme has the advantages of theoretical rigor and high programming efficiency and that the efficiency of cross-face adjacent cell searching is 242.9 times that of a similar scheme. Moreover, the proposed complex radix number representation is an ideal formalized description tool for grid systems. The research ideas introduced herein can be used to create a universal theoretical framework for DGGSs.
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Acknowledgements
The authors would like to thank three anonymous reviewers for their constructive and insightful comments which helped to significantly improve this manuscript. The authors would also like to thank associate professor TONG Xiaochong at the Institute of Surveying and Mapping, Information Engineering University, China for generous providing C++ source code and test program of HQBS. This research was supported by the National Natural Science Foundation of China (Grant No. 41671410), the Postdoctoral Science Foundation of China (Grant No. 2013T60161), and the Excellent Young Scholar Foundation of Information Engineering University (Grant No. 2016610802).
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Ben, J., Li, Y., Zhou, C. et al. Algebraic encoding scheme for aperture 3 hexagonal discrete global grid system. Sci. China Earth Sci. 61, 215–227 (2018). https://doi.org/10.1007/s11430-017-9111-y
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DOI: https://doi.org/10.1007/s11430-017-9111-y