# General Method for Extending Discrete Global Grid Systems to Three Dimensions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. DGGS Overview

#### 3.1. Initial Discretization

#### 3.2. Refinement

#### 3.3. Projection and Inverse Projection

#### 3.4. Indexing Scheme

## 4. Prismatoid Grid Generation and Refinement

#### 4.1. Initial 3D Discretization

#### 4.2. Regular Prismatoid Refinement

#### 4.3. Semiregular Prismatoid Refinement

#### 4.4. Other Surface Refinement Factors

#### 4.5. Cell Aspect Ratio

## 5. Encoding and Decoding for 3D DGGS

#### 5.1. Radial Mapping

#### 5.2. Layer Parameterization

## 6. Indexing Operations

## 7. Results and Evaluation

#### 7.1. Aircraft and Satellite Paths

#### 7.2. Urban Planning

#### 7.3. Atmospheric Properties

#### 7.4. Discussion

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GIS | Geographic Information System |

DGGS | Discrete Global Grid System |

3D DGGS | Three-dimensional Discrete Global Grid System |

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**Figure 1.**Overview of how the input Discrete Global Grid System (DGGS) is used to define the geometry of the three-dimensional (3D) DGGS. The faces of the initial polyhedron of the input DGGS are first extruded to create the prismatoid base cells of the 3D DGGS. For refinement, the bases of these prismatoid cells are refined using the same refinement scheme as the input DGGS and combined with a radial refinement. A semiregular refinement is suggested; however, regular refinement is also possible. A more detailed description of these processes is provided in Section 4.

**Figure 2.**The process of encoding (left) and decoding (right) with a DGGS. Encoding: features are projected from the Earth domain to the polyhedral domain of the grid; the set of planar cells associated with the feature are obtained—each of which has a unique index. Decoding: a set of cell indices corresponds to cell geometry on the polyhedral grid; cells in the polyhedral domain are inverse projected back to the Earth domain. The process for a 3D DGGS is similar, except features and cells have associated altitude(s).

**Figure 3.**The radius of a polyhedron is defined as the radius of its circumscribing sphere. In this figure, all points on a solid line of a given colour have the same radius as they all have the same circumscribing sphere (i.e., lie on the same polyhedron). The corresponding spheres are shown in the same colours but with a dashed line.

**Figure 5.**Semiregular prismatoid refinement as applied to a single pyramid cell (

**a**) once and (

**b**) twice. In (

**b**), the number of times the surface refinement has been applied (${k}_{s}$) is shown for each layer; see how the radial refinements of central layers separate regions of the grid with different values of ${k}_{s}$.

**Figure 6.**Three levels of successive applications of regular (top) and semiregular (bottom) prismatoid refinement. Only the side of a single pyramid starting cell is shown to highlight the behaviour of the two schemes at different radii. Note how cells degenerate towards the centre with regular refinement, whereas this issue is not present in the semiregular scheme.

**Figure 7.**Semiregular prismatoid refinement applied to a hexagonal base cell using a 1:3 surface refinement scheme (

**a**) once and (

**c**) twice; (

**b**) is the same image as (

**c**), but with the surface cells of the previous refinement level shown for reference. Note that in the second level of refinement, normal layers have two radial refinements as opposed to one.

**Figure 8.**A demonstration of how refinement can be modified to affect the aspect ratio of cells. All figures show a starting pyramid cell from a grid with 200 cells in its initial discretization. Central layer with five extra radial splits (a = 3 to get x = 5) at (

**a**) one level of refinement and (

**b**) three levels. Central layer with two applications of the surface refinement scheme (a = 1/8 to get w = 2) at (

**c**) one level of refinement and (

**d**) three levels.

**Figure 9.**Indices from the input DGGS identify cells in the 3D DGGS with the same bases but different radii. Likewise, each layer of a 3D DGGS contains all cells with the same radii but different bases. Thus, these two pieces of information combined are sufficient to identify any cell in the 3D DGGS uniquely. Here we show a single cell—highlighted in purple—along with all cells that share a surface index (red) and a layer index (blue).

**Figure 10.**Pipeline for splitting encoding and decoding into surface (top) and radial (bottom) components. These two components are entirely independent except for the surface resolution (${k}_{s}$) of the layer being provided to the surface component during encoding. Standard notation for latitude and longitude ($\varphi $ and $\lambda $) is used.

**Figure 11.**Radial splits of central layers divide the grid into regular regions that represent spherical shells. At refinement level k, there are $k-1$ shells and the central layer. These shells are similar, but have a different number of cells and should, therefore, have a mapped volume proportional to the number of cells they contain.

**Figure 12.**Our three radial mappings applied to the same grid, along with a normalization polyhedral projection. Left: first mapping with linear interpolation. Right: second mapping with cubic interpolation. Middle: third mapping with quadratic interpolation. Note how cells in the right mapping are stretched and squashed in order to preserve their volume better. This effect is also present in the middle mapping, but to a lesser extent.

**Figure 13.**Flight paths and satellite orbits rasterized in a 3D DGGS. All paths and orbits are rasterized at the same grid resolution of (

**a**) 7, (

**b**) 10, and (

**c**) 13.

**Figure 14.**A collection of buildings from the University of Calgary rasterized in a 3D DGGS at resolution 21. In (

**a**), both interior and boundary cells are shown, whereas (

**b**,

**c**) show only boundary and interior cells, respectively.

**Figure 15.**Vertical wind speed (mbar/s) sampled into a 3D DGGS at resolution 9. Negative velocity, which corresponds to an upward wind, is shown in red. Positive velocity, which corresponds to a downward wind, is shown in blue. Velocities with a magnitude less than 0.25 mbars/s are not shown to reduce clutter and highlight regions with the highest speeds. In (

**a**,

**c**), the altitude of cells is scaled by a factor of 15 to better show changes in altitude. In (

**b**), altitude is shown at its true scale.

**Table 1.**Our proposed layering sequences for different surface refinement factors. Each number in the sequence is how many layers normal layers should split into at the corresponding level of refinement. An overline indicates that the sequence repeats indefinitely. For semiregular prismatoid refinement, since the initial discretization has no normal layers, these sequences are used starting with the second level of refinement

Surface Refinement Factor | Layering Sequence |
---|---|

1:2 | $\overline{2,1}$ |

1:3 | $\overline{3,1}$ |

1:4 | $\overline{2,2}$ |

1:5 | 2,3,2,2,2,3,2,2,2,3,2,2,3... |

1:6 | $\overline{3,2}$ |

1:7 | 3,2,3,3,2,3,3,2,3,3,3,2,3... |

1:8 | $\overline{4,2}$ |

1:9 | $\overline{3,3}$ |

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**MDPI and ACS Style**

Ulmer, B.; Hall, J.; Samavati, F.
General Method for Extending Discrete Global Grid Systems to Three Dimensions. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 233.
https://doi.org/10.3390/ijgi9040233

**AMA Style**

Ulmer B, Hall J, Samavati F.
General Method for Extending Discrete Global Grid Systems to Three Dimensions. *ISPRS International Journal of Geo-Information*. 2020; 9(4):233.
https://doi.org/10.3390/ijgi9040233

**Chicago/Turabian Style**

Ulmer, Benjamin, John Hall, and Faramarz Samavati.
2020. "General Method for Extending Discrete Global Grid Systems to Three Dimensions" *ISPRS International Journal of Geo-Information* 9, no. 4: 233.
https://doi.org/10.3390/ijgi9040233