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Adiscrete global grid (DGG) is amosaic that covers the entire Earth's surface.Mathematically it is aspace partitioning: it consists of a set of non-empty regions that form apartition of the Earth's surface.^[1] In a usual grid-modeling strategy, to simplify position calculations, each region is represented by a point, abstracting the grid as a set of region-points. Each region or region-point in the grid is called acell.

When each cell of a grid is subject to arecursive partition, resulting in a "series of discrete global grids with progressively finer resolution",^[2] forming a hierarchical grid, it is called ahierarchical DGG (sometimes "global hierarchical tessellation"^[3]or "DGG system").

Discrete global grids are used as the geometric basis for the building of geospatialdata structures. Each cell isrelated with data objects or values, or (in the hierarchical case) may be associated with other cells.DGGs have been proposed for use in a wide range of geospatial applications, including vector and raster location representation, data fusion, and spatial databases.^[1]

The most usual grids are forhorizontal position representation, using a standarddatum, likeWGS84. In this context, it is common also to use a specific DGG as foundation forgeocoding standardization.

In the context of aspatial index, a DGG can assign unique identifiers to each grid cell, using it for spatial indexing purposes, ingeodatabases or forgeocoding.

The "globe", in the DGG concept, has no strict semantics, but in geodesy a so-called "grid reference system" is a grid that divides space with precise positions relative to adatum, that is an approximated "standard model of theGeoid". So, in the role of Geoid, the "globe" covered by a DGG can be any of the following objects:

• Thetopographical surface of the Earth, when each cell of the grid has its surface-position coordinates and the elevation in relation to thestandard Geoid. Example: grid with coordinates (φ,λ,z) wherez is the elevation.
• AstandardGeoid surface. The z coordinate is zero for all grid, thus can be omitted, (φ,λ).
  Ancient standards, before 1687 (the Newton's Principia publication), used a "reference sphere"; in nowadays the Geoid is mathematically abstracted asreference ellipsoid.
  • Asimplified Geoid: sometimes an old geodesic standard (e.g.SAD69) or a non-geodesic surface (e. g. perfectly spherical surface) must be adopted, and will be covered by the grid. In this case, cells must be labeled with non-ambiguous way,(φ',λ'), and the transformation (φ,λ) ⟾ (φ′,λ′) must be known.
• Aprojection surface. Typically the geographic coordinates (φ,λ) are projected (with some distortion) onto the 2D mapping plane with 2D Cartesian coordinates (x, y).

As a global modeling process, modern DGGs, when including projection process, tend to avoid surfaces like cylinder or a conic solids that result in discontinuities and indexing problems.Regular polyhedra and other topological equivalents of sphere led to the most promising known options to be covered by DGGs,^[1] because "spherical projections preserve thecorrect topology of the Earth – there are no singularities or discontinuities to deal with".^[4]

When working with a DGG, it is important to specify which of these options was adopted. So, the characterization of thereference model of the globe of a DGG can be summarized by:

• The recoveredobject: the object type in the role of globe. If there is no projection, the object covered by the grid is the Geoid, the Earth or a sphere; else, it is the geometry class of the projection surface (e.g. a cylinder, a cube or a cone).
• Projection type: absent (no projection) or present. When present, its characterization can be summarized by theprojection's goal property (e.g. equal-area, conformal, etc.) and the class of the corrective function (e.g. trigonometric, linear, quadratic, etc.).

NOTE: When the DGG is covering a projection surface, in a context ofdata provenance, the metadata about its reference Geoid is also important — typically specifying itsISO 19111's CRS value, with no confusion with the projection surface.

The main distinguishing feature to classify or compare DGGs is the use or not of hierarchical grid structures:

• Inhierarchical reference systems each cell is a "box reference" to a subset of cells, and cell identifiers can express this hierarchy in its numbering logic or structure.
• Innon-hierarchical reference systems each cell have a distinct identifier and represents a fixed-scale region of the space. The discretization of theLatitude/Longitude system is the most popular, and the standard reference for conversions.

Other usual criteria to classify a DGG are tile-shape and granularity (grid resolution):

• Tile regularity and shape: there are regular, semi-regular orirregular grid. As in generictilings by regular polygons, is possible to tiling with regular face (like wall tiles can be rectangular, triangular, hexagonal, etc.), or with same face type but changing its size or angles, resulting in semi-regular shapes.
  Uniformity of shape and regularity of metrics provide better grid-indexing algorithms. Although it has less practical use, totally irregular grids are possible, such in aVoronoi coverage.
• Fine or coarsegranulation (cell size): modern DGGs are parametrizable in its grid resolution, so, it is a characteristic of the final DGG instance, but not useful to classify DGGs, except when the DGG-type must use a specific resolution or have a discretization limit. A "fine" granulation grid is non-limited and "coarse" refers to drastic limitation. Historically the main limitations are related to digital/analogic media, the compression/expanded representations of the grid in a database, and the memory limitations to store the grid. When a quantitative characterization is necessary, the average area of the grid cells or average distance between cell centers can be adopted.

The most common class of discrete global grids are those that place cell center points on longitude/latitude meridians and parallels, or which use the longitude/latitude meridians and parallels to form the boundaries of rectangular cells. Examples of such grids, all based on latitude/longitude:

UTM zones: Divides the Earth into sixty (strip) zones, each being a six-degree band of longitude. In digital media removes overlapping zone. Use secant transverse Mercator projection in each zone. Define 60 secant cylinders, 1 per zone. TheUTM zones was enhanced byMilitary Grid Reference System (MGRS), by addition of theLatitude bands.
inception: 1940s	covered object: cylinder (60 options)	projection: UTM or latlong	irregular tiles: polygonal strips	granularity: coarse

(modern) UTM –Universal Transverse Mercator: Is a discretization of the continuous UTM grid, with a kind of 2-level hierarchy, where the first level (coarse grain) correspond to the "UTM zones with latitude bands" (theMGRS), use the same 60 cylinders as reference-projection objects. Each fine-grain cell is designated by a structured ID composed by "grid zone designator", "the 100,000-meter square identifier" and "numerical location". The grid resolution is a direct function of the number of digits in the coordinates, that is also standardized. For instance the cell17N 630084 4833438 is a ~10mx10m square. PS: this standard use 60 distinct cylinders for projections. There are also "Regional Transverse Mercator" (RTM or UTM Regional) and "Local Transverse Mercator" (LTM or UTM Local) standards, with more specific cylinders, for better fit and precision at the point of interest.
inception: 1950s	covered object: cylinder (60 options)	projection: UTM	rectangular tiles: equal-angle (conformal)	granularity: fine

ISO 6709: Discretizes the traditional "graticule" representation and the modern numeric-coordinate cell-based locations. The granularity is fixed by a simple convention of the numeric representation, e. g. one-degree graticule, .01 degree graticule, etc. and it results in non-equal-area cells over the grid. The shape of the cells are rectangular except in the poles, where they are triangular. The numeric representation is standardized by two main conventions: degrees (Annex D) and decimal (Annex F). The grid resolution is controlled by the number of digits (Annex H).
inception: 1983	covered object: Geoid (anyISO 19111's CRS)	projection: none	rectangular tiles: uniform spheroidal shape	granularity: fine

PrimaryDEM (TIN DEM): A vector-based triangular irregular network (TIN) — the TIN DEM dataset is also referred to as a primary (measured) DEM. Many DEM are created on a grid of points placed at a regular angular increments of latitude and longitude. Examples include theGlobal 30 Arc-Second Elevation Dataset (GTOPO30).[5] and the Global Multi-resolution Terrain Elevation Data 2010 (GMTED2010).[6]Triangulated irregular network is a representation of a continuous surface consisting entirely of triangular facets.
inception: 1970s	covered object: terrain	projection: none	triangular non-uniform tiles: parametrized (vectorial)	granularity: fine

The right aside illustration show 3 boundary maps of the coast of Great Britain. The first map was covered by a grid-level-0 with 150 km size cells. Only a grey cell in the center, with no need of zoom for detail, remains level-0; all other cells of the second map was partitioned into four-cells-grid (grid-level-1), each with 75 km. In the third map 12 cells level-1 remains as grey, all other was partitioned again, each level-1-cell transformed into a level-2-grid.
Examples of DGGs that use such recursive process, generating hierarchical grids, include:

Hierarchical Equal Area isoLatitude Pixelization (HEALPix): HEALPix has equal area quadrilateral-shaped cells and was originally developed for use with full-sky astrophysical data sets.[16] The usual projection is "H?4, K?3 HEALPix projection".Main advantage, comparing with others of same indexing niche as S2, "is suitable for calculations involving spherical harmonics".[17]
inception: 2006	covered object: Geoid	projection: (K,H) parametrized HEALPix projection	quadrilateral tiles: uniform area-preserved	granularity: fine

Hierarchical Triangular Mesh (HTM): Developed in 2003...2007, HTM "is a multi level, recursive decomposition of the sphere. It start with an octahedron, let this be level 0. As you project the edges of the octahedron onto the (unit) sphere creates 8 spherical triangles, 4 on the Northern and 4 on the Southern hemispheres".[18] Then, each triangle is refined into 4 subtriangles (1-to-4 split). The first public operational version seems[19] the HTM-v2 in 2004.
inception: 2004	covered object: Geoid	projection: none	triangular tiles: spherical equilateres	granularity: fine

Geohash: Latitude and longitude are merged, enterlacing bits in the joined number. The binary result is represented with base32, offering a compact human-readable code. When used asspatial index, corresponds to aZ-order curve. There are some variants likeGeohash-36.
inception: 2008	covered object: Geoid	projection: none	semi-regular tiles: rectangular	granularity: fine

S2 / S2Region: The "S2 Grid System" is part of the "S2 Geometry Library"[20] (the name is derived from the mathematical notation for then-sphere,Sn). It implements an index system based oncube projection and the space-fillingHilbert curve, developed atGoogle.[21][22] The S2Region of S2 is the most general representation of its cells, where cell-position and metric (e.g. area) can be calculated. Each S2Region is a subgrid, resulting in a hierarchy limited to 31 levels. At level30 resolution is estimated[23] in 1 cm2, at level0 is 85011012 km2. The cell-identifier of the hierarchical grid of a cube face (6 faces) have and ID of 60 bits (so "every cm2 on Earth can be represented using a 64-bit integer).
inception: 2015	covered object: cube	projection: spherical projections in each cube face using quadratic function	semi-regular tiles: quadrilateral projections	granularity: fine

There is a class of hierarchical DGG's named by theOpen Geospatial Consortium (OGC) as "discrete global grid systems" (DGGS), that has to satisfy 18 requirements. Among them, what best distinguishes this class from other hierarchical DGGs, is the Requirement-8,"For each successive level of grid refinement, and for each cell geometry, (...) Cells that are equal area (...) within the specified level of precision".^[27]

A DGGS is designed as a framework for information as distinct from conventionalcoordinate reference systems originally designed for navigation. For a grid-based global spatial information framework to operate effectively as an analytical system it should be constructed using cells that represent the surface of the Earth uniformly.^[27] The DGGS standard includes in its requirements a set of functions and operations that the framework has to offer.

All DGGS's level-0 cells are equal area faces of aregular polyhedron.

The standard defines therequirements of a hierarchical DGG, including how to operate the grid. Any DGG that satisfies these requirements can be named DGGS."A DGGS specification SHALL include a DGGS Reference Frame and the associated Functional Algorithms as defined by the DGGS Core Conceptual Data Model".^[28]

For an Earth grid system to be compliant with this Abstract Specification it must define a hierarchical tessellation of equal area cells that both partition the entire Earth at multiple levels of granularity and provide a global spatial reference frame. The system must also include encoding methods to: address each cell; assign quantized data to cells; and perform algebraic operations on the cells and the data assigned to them. Main concepts of the DGGS Core Conceptual Data Model:

1. reference frame elements, and,
2. functional algorithm elements; comprising:
   1. quantization operations,
   2. algebraic operations, and
   3. interoperability operations.

There are many DGGs because there are many representational, optimization and modeling alternatives. All DGG grid is a composition of its cells, and, in the Hierarchical DGG each cell uses a new grid over its local region.

The illustration is not adequate toTIN DEM cases and similar "raw data" structures, where the database not use the cell concept (that geometrically is the triangular region), but nodes and edges: each node is an elevation and each edge is the distance between two nodes.

In general, each cell of the DGG is identified by the coordinates of its region-point (illustrated as the centralPoint of a database representation). It is also possible, with loss of functionality, to use a "free identifier", that is, any unique number or unique symbolic label per cell, thecell ID. The ID is usually used as spatial index (such as internalQuadtree ork-d tree), but is also possible to transform ID into a human-readable label forgeocoding applications.

Modern databases (e.g. using S2 grid) use also multiple representations for the same data, offering both, a grid (or cell region) based in the Geoid and a grid-based in the projection.

Discrete global grids with cell regions defined by parallels and meridians oflatitude/longitude have been used since the earliest days of globalgeospatial computing. Before it, the discretization of continuous coordinates for practical purposes, with paper maps, occurred only with low granularity. Perhaps the most representative and main example of DGG of this pre-digital era was the 1940s militaryUTM DGGs, with finer granulated cell identification forgeocoding purposes. Similarly somehierarchical grid exists before geospatial computing, but only in coarse granulation.

A global surface is not required for use on daily geographical maps, and the memory was very expensive before the 2000s, to put all planetary data into the same computer. The first digital global grids were used for data processing of the satellite images and global (climatic andoceanographic) fluid dynamics modeling.

The first published references tohierarchical geodesic DGG systems are to systems developed for atmospheric modeling and published in 1968. These systems have hexagonal cell regions created on the surface of a sphericalicosahedron.^[29][30]

The spatial hierarchical grids were subject to more intensive studies in the 1980s,^[31] when main structures, asQuadtree, were adapted in image indexing and databases.

While specific instances of these grids have been in use for decades, the termdiscrete global grids was coined by researchers atOregon State University in 1997^[2] to describe the class of all such entities.

... OGC standardization in 2017...

The evaluation discrete global grid consists of many aspects, including area, shape, compactness, etc.Evaluation methods formap projection, such asTissot's indicatrix, are also suitable for evaluating map projection-based discrete global grid.

In addition, averaged ratio between complementary profiles (AveRaComp)^[32] gives a good evaluation of shape distortions for quadrilateral-shaped discrete global grid.

Database development-choices and adaptations are oriented by practical demands for greater performance, reliability or precision. The best choices are being selected and adapted to necessities, propitiating the evolution of the DGG architectures. Examples of this evolution process: from non-hierarchical to hierarchical DGGs; from the use of Z-curve indexes (anaive algorithm based in digits-interlacing), used by Geohash, to Hilbert-curve indexes, used in modern optimizations, like S2.

In general each cell of the grid is identified by the coordinates of its region-point, but it is also possible to simplify the coordinate syntax and semantics, to obtain an identifier, as in a classicalphanumeric grids — and find the coordinates of a region-point from its identifier. Small and fast coordinate representations is a goal in the cell-ID implementations, for any DGG solutions.

There is no loss of functionality when using a "free identifier" instead of a coordinate, that is, any unique number (or unique symbolic label) per region-point, thecell ID. So, to transform a coordinate into a human-readable label, and/or compressing the length of the label, is an additional step in the grid representation. This representation is namedgeocode.

Some popular "global place codes" asISO 3166-1 alpha-2 for administrative regions orLonghurst code for ecological regions of the globe, arepartial in globe's coverage. By other hand, any set of cell-identifiers of a specific DGG can be used as "full-coverage place codes". Each different set of IDs, when used as a standard for data interchange purposes, are named "geocoding system".

There are many ways to represent the value of a cell identifier (cell-ID) of a grid: structured or monolithic, binary or not, human-readable or not. Supposing a map feature, like theSingapore's Merlion fountaine (~5m scale feature), represented by itsminimum bounding cell or a center-point-cell, thecell ID will be:

All these geocodes represents the same position in the globe, with similar precision, but differ instring-length, separators-use and alphabet (non-separator characters). In some cases the "original DGG" representation can be used. The variants are minor changes, affecting only final representation, for example the base of the numeric representation, or interlacing parts of the structured into only one number or code representation. The most popular variants are used for geocoding applications.

DGGs and its variants, withhuman-readable cell-identifiers, has been used asde facto standard foralphanumeric grids. It is not limited to alphanumeric symbols, but "alphanumeric" is the most usual term.

Geocodes are notations for locations, and in a DGG context, notations to express grid cell IDs. There are a continuous evolution in digital standards and DGGs, so a continuous change in the popularity of each geocoding convention in the last years. Broader adoption also depends on country's government adoption, use in popular mapping platforms, and many other factors.

Examples used in the following list are about "minor grid cell" containing theWashington obelisk,38° 53′ 22.11″ N, 77° 2′ 6.88″ W.

Other documented systems:

• Grid reference
• Geodesic grid
• List of geocoding systems
• Military Grid Reference System

• OGC DGGS Standards Working Group
• Discrete Global Grids page at the Computer Science department at Southern Oregon University
• BUGS climate modelArchived 2006-12-15 at theWayback Machine page on geodesic grids
• Research Institute for World Grid squares page on World Grid Squares
• Cubic Postcode a valid protocol for an international postcode system using a grid of cubic metres
• Earth grid models for exhibition use. 3D-printable models of some Earth grids.

• Geographic coordinate systems

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