Divide and conquer – often true for Geospatial Data
Recently, I was asked to review a paper that dealt with the cartographic visualization of a global dataset on a digital globe. The dataset consisted of a large number of points, each marking the occurrence of an event. The authors had the sensible idea of aggregating these points into cells of equal size. Each cell (more precisely, its centroid) would then be visualized based on the number of events it contained. This approach, known as tessellation — dividing space into regular tiles that serve as aggregation units for people, trees, accidents, and so on — is widely known and used. In the past, these tiles were almost always square grid cells, but hexagons have become increasingly common. Hexagons offer one major advantage: the centroids of all neighbouring cells are equidistant from the centre of the reference cell. However, they have a significant drawback in hierarchical systems with multiple spatial resolution levels: higher-order hexagons do not align neatly with lower-order hexagons. Squares, in contrast, have no such issue—if each square cell is divided into four or nine smaller cells, each “child” cell can be uniquely assigned to a “parent” cell at a higher aggregation level.
Hierarchical Tessellations: Squares vs. Hexagons. As the comparison between square grid variants shows, the resolution increase per hierarchical level can differ depending on the aperture of the system—that is, the area ratio between a tile and its corresponding tile at the next coarser resolution level. Thus, the square grid on the left has aperture 4, the grid in the middle aperture 9 and the hexagonal grid aperture 7.
Lets get back to the manuscript: The authors described their use of a hexagonal grid in a Web Mercator projection to assign each hexagon the number of Point events within it. But then I read that each cell covers an area of 30,000 square kilometres — and paused. A regular grid on a Web Mercator projection, with its extreme area distortions at higher latitudes, cannot produce equal-area cells! On the rotatable digital globe accompanying the paper, this became visually obvious: around Stockholm or St. Petersburg, the distance between cell centroids shrank to roughly half of the corresponding cells at the equator. Consequently, the actual area covered by a cell near St. Petersburg was only about a quarter of that of a near-equatorial cell. This is far from ideal for a count grid intended to enable meaningful comparisons of event occurrences across the globe.
A grid based on a Web Mercator projection (left). On the right, a representation of this grid on a digital globe, which has been rotated at a constant scale (or viewing distance) so that the viewer is positioned centrally above the respective section (Scandinavia at the top, Congo at the bottom).
An equitable partitioning of the Earth?
Defining a global count grid with equal-area cells is challenging. Geographic coordinates are unsuitable because the areas formed by latitude and longitude lines (e.g., 1° × 1° cells) shrink toward the poles. Datasets structured this way, such as the SRTM elevation model, also have the disadvantage of trapezoidal grid cells of variable width, which can be problematic for GIS analyses that assume square cells. One possible solution is to use an equal-area projection (e.g., Equal Earth) as a basis and overlay a regular, potentially hierarchical grid. In this case, every grid cell would represent the same surface area on Earth. However, the shape of the area each cell covers would vary dramatically across the globe. While this is acceptable for purely comparative global occurrence analyses, it creates serious issues for analyses that assume uniform distances between grid cells — for example, calculations assuming all diagonal neighbours are equidistant. In such cases, results would be distorted.
A counting grid superimposed on an equal-area projection (Equal Earth). Although the area of the Earth’s surface covered by each cell is the same, the shape of the covered area would vary greatly, as shown by the orange Tissot indicatrix. This depicts real (i.e. on the surface of the sphere) circles of equal size on the map image. Just as the ellipses resulting from the projection deviate significantly from the circular shape on the map, the grid cells of the map would deviate from the square shape in nature. While the deviation is still limited in the centre of the projection, it increases significantly towards the edges. (Source: modified from Justin Kunimune, CC0, via Wikimedia Commons)
What if, instead of projecting the Earth’s surface onto a plane, we project it onto a polyhedron — a three-dimensional shape with flat faces, such as a cube? Since the facets of such polyhedra approximate the curvature of the Earth more closely than a single plane, the distortions introduced by “flattening” would be far less severe.
The more spherical the polyhedron (from top: octahedron, tetrahedron, slightly damaged icosahedron), the less distortion there is in the projection (…and the easier they are to juggle ;-)).
DGGS – The Earth in Trillions of Pieces
Equal-area Discrete Global Grid Systems (DGGS) take exactly this approach. The Earth’s surface is projected area-preserving onto a Platonic solid such as a cube (six square faces) or an icosahedron (20 equilateral triangles). Based on this projection, a hierarchical grid is defined in which every location on Earth is unambiguously assigned to a cell at any chosen resolution level. Unlike traditional map visualizations, which require “cutting and unfolding” the polyhedral faces, DGGS calculations operate seamlessly across polyhedron edges, supporting spatial analyses on a global scale without discontinuities.
The idea of projecting the Earth onto a Platonic solid to minimise distortion is not new: in 1954, Buckminster Fuller and Shoji Sadao presented this (albeit not area-preserving) projection of the Earth onto an icosahedron, which was then “cut and unfolded” so that all the cuts lay in the oceans. (Source: Justin Kunimune, CC0, via Wikimedia Commons)
In short, DGGS offer seamless, global grids in hierarchically organised scale levels. In contrast to point-oriented, i.e. continuous latitude and longitude specifications of traditional global reference systems, DGGS discretise the Earth into hierarchically organised, flat tiles. This offers particular advantages when integrating and querying very large, global data sets. Instead of performing complex, spherical calculations in corresponding vector data sets, these are first converted into a DGGS, where they are aggregated spatially at suitable (or all) resolution levels. Queries and analyses of the now grid-based data can be performed very efficiently thanks to spatial indexing of the cells. One of the most prominent implementations is H3, developed by Uber to analyse global demand and supply for its ride-hailing services and to enable dynamic, location-based pricing.
The species and four specimens
In 2017, the Open Geospatial Consortium (OGC) published the first abstract specification for DGGS, i.e. a technology-independent description of the underlying concepts. While this first version included the hierarchical partitioning of the entire Earth and tiles of equal size within a hierarchical level as constitutive elements of a DGGS, this is no longer the case in the current version 2.0 from 2021. Area-preserving DGGS are described there as a subcategory of DGGS. This is probably because popular DGGS such as the aforementioned H3 (https://h3geo.org/) or Google’s ‘S2 Geometry’ (https://s2geometry.io/) exhibit slight variations in the area of the hexagonal (H3) and square (S2) tiles.
H3 (like S2) is open source, has its own API that can be addressed in several languages, and is now so widespread that classic GI systems such as QGIS or ArcGIS can generate the specific tile structure. It is based primarily on hexagons (Aperture 7) in 15 resolution levels, with a cell in the highest resolution covering an area of less than one square metre. ‘Primarily’ because the tessellation of an icosahedron surface is not possible with hexagons alone but always requires 12 pentagons as well. In H3, these pentagons are typically placed over oceans, minimizing their impact.
A classic football consists of 12 pentagons and 20 hexagons. In H3, 12 pentagons are also required in each resolution, regardless of whether 110 (highest aggregation level) or 569,707,381,193,150 hexagons (highest resolution) are added. The pentagons are significantly smaller, but according to the OGC specification, area equality (for area-preserving DGGS, which does not apply to H3 anyway) is only required within the same tile geometry.
Examples of area-preserving DGGS established in science are ISEA3H (Sahr et al. 2003) and HEALPix (Górski et al. 1998), which originated in astronomy. ISEA3H uses area-preserving hexagons projected onto an icosahedron (Aperture 3), while HEALPix uses square to diamond-shaped tiles projected onto a rhombic dodecahedron. Due to their area-preserving nature, both systems are suitable for modelling and statistical analysis of global biological and geoscientific data, such as those found in habitat or climate modelling. For aggregating the point dataset discussed at the beginning of this article, these systems would have been excellent choices.
Górski, K. M., Hivon, E., & Wandelt, B. D. (1998) Analysis issues for large CMB data sets. https://doi.org/10.48550/arXiv.astro-ph/9812350.
Li, M., Stefanakis, E. (2020) Geospatial Operations of Discrete Global Grid Systems—a Comparison with Traditional GIS. Journal of Geovisualization and Spatial Analysis 4, 26. https://doi.org/10.1007/s41651-020-00066-3
Mechenich, M.F., Žliobaitė, I. (2023) Eco-ISEA3H, a machine learning ready spatial database for ecometric and species distribution modeling. Sci Data 10, 77. https://doi.org/10.1038/s41597-023-01966-x
Open Geospatial Consortium (2021) Topic 21 – Discrete Global Grid Systems – Part 1 Core Reference system and Operations and Equal Area Earth Reference System. Vers. 2.0 https://docs.ogc.org/as/20-040r3/20-040r3.html
Sahr, K., D. White & A. J. Kimerling (2003) Geodesic Discrete Global Grid Systems, Cartography and Geographic Information Science, 30:2, 121-134, DOI: 10.1559/152304003100011090
