A New Earth System Spatial Grid Extending the Great Circle Arc QTM: The Spherical Geodesic Degenerate Octree Grid

Abstract

An Earth system spatial grid (ESSG) is an extension of a discrete global grid system (DGGS) in the radial direction. It is an important tool for organizing, representing, simulating, analyzing, sharing, and visualizing spatial data. The existing ESSGs suffer from complex spatial relationships and significant geometric distortion. To mitigate these problems, a spherical geodesic degenerate octree grid (SGDOG) and its encoding and decoding schemes are proposed in this paper. The SGDOG extends the great circle arc QTM in the radial direction and adopts different levels of the great circle arc QTM at different radial depths. The subdivision of SGDOG is simple and clear, and has multi-level characteristics. The experimental results demonstrate that the SGDOG has advantages of simple spatial relationships, convergent volume distortion, and real-time encoding and decoding. The SGDOG has the potential to organize and manage global spatial data and perform large-scale visual analysis of the Earth system.

1. Introduction

Scientific and technological advances in recent years have enabled Earth research to extend from surface studies to interior and exterior exploration. Various types of sensors, such as satellite-based remote sensors, atmospheric monitoring sensors, geodetic sensors, and underground detection devices, collect data from high altitudes and underground environments [1,2]. The acquired data are utilized for applications such as satellite trajectory tracking [3], atmospheric circulation analysis [4], crustal movement monitoring [5], and large-scale urban modeling [6]. The traditional method of processing these data is to project the Earth’s surface onto a plane. The geometric distortion and map edges will affect the accuracy of certain analyses [7]. The increasing amount of spatial data poses challenges to traditional geographic information technology and tools for organizing and processing data [8,9,10,11]. A data model suitable for organizing and processing the large amount of spatial data in the global scale is needed.

The discrete global grid system (DGGS) framework is a spherical surface grid theory and an innovative type of geographic information system (GIS) and method of spatial data processing [12,13]. DGGSs employ a discrete and hierarchical structure by dividing the Earth’s surface into regular and non-overlapping cells with unique identifiers [14] to establish an organized and efficient spatial data structure.

An Earth system spatial grid (ESSG) [15], also called a 3D-DGGS by some scholars [16,17], extends a DGGS in the radial direction. ESSGs are important tools for organizing, representing, simulating, analyzing, sharing, and visualizing spatial data in 3D GIS, and can support the integration of multi-source spatial data [18]. The ESSG framework has been applied to match crustal data to establish crustal models [16,19], represent Earth’s structure to calculate mantle P-wave velocity [20,21], estimate feature volumes in urban planning and management [7], discretely carry meteorological data for the differential operation of atmospheric dynamics equations and temperature field modeling [19], and for many other purposes. The research on ESSGs mainly focuses on 3D spherical latitude–longitude grids and 3D spherical regular polyhedron grids.

A 3D spherical latitude–longitude grid divides the sphere using latitude and longitude lines and spherical surfaces at specific intervals, offering the advantage of orthogonal grid boundaries that coincide with the latitude and longitude lines. Its disadvantages are as follows: (1) it is difficult to extend to an ellipsoid since the normals of points on the ellipsoid, except for the poles and the points on equator, do not pass through the center of the ellipsoid [22]; and (2) the shapes and sizes of cells vary significantly when the latitude–longitude grid is extended in the radial direction, such as GeoSOT-3D [23]. A Yin–Yang grid [24] has been proposed to avoid the convergence of the latitude–longitude grid. However, the boundaries of the Yin and Yang regions overlap, leading to a non-unique mapping between physical space and grid space. The ambiguous expression of spatial and attribute information in the Yin–Yang boundary region contradicts the unified expression, modeling, and analysis of the sphere. To address the issue of the cell sizes decreasing toward the poles and the center of the sphere, the spherical degenerated octree grid (SDOG) [25] has been proposed, along with a corresponding multi-level degenerated Z-curve filling coding (MDZ coding). The SDOG extends the degenerated quadtree grid (DQG) [26,27] in the radial direction and is applied to the stereoscopic modeling of large-scale spatial objects [28]. Although the SDOG maintains approximate cell volumes, it features complex cell shapes and spatial relationships.

A 3D spherical regular polyhedron grid extends the spherical surface regular polyhedron grid in the radial direction. The spherical surface regular polyhedron grid is generally divided based on five regular polyhedra: the regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron, and regular icosahedron [29]. Their cell shapes are primarily triangles [30], quadrilaterals [31], and hexagons [32,33]. The hexagonal hierarchical geospatial indexing system (H3) is a hexagonal grid system with a seven-aperture hierarchical subdivision, based on a regular icosahedron [33]. Compared with quadrilateral and triangular grids, hexagonal grids resemble circles more closely in shape, offering uniform distances between adjacent cells and achieving the optimal perimeter-to-area ratio. However, the hierarchical subdivision strategy for hexagonal grids is more complex than that for quadrangular and triangular grids. The quaternary triangular mesh (QTM) based on the triangular subdivision in a regular octahedron, has advantages of well-defined vertex positioning, simple structure, moderate geometric distortion, and extendibility to ellipsoid form [34]. Thus, it has been widely applied in global spatial data extraction and multi-resolution management [35], spatial hierarchical indexing [36], global navigation [37], global DEM generation [38], and remote sensing data organization [39]. However, compared to grid systems based on icosahedra, the geometric distortion of QTM is larger [34]. There are two methods to determine the midpoints of the spherical triangle edges in the QTM: (1) taking the mid-value of the longitude and latitude [40,41], and (2) taking the midpoint of the great circle arc [42]. Except for midpoints on longitude lines or the equator, the midpoints generated by taking the mid-values of the longitude and latitude are not on the great circle of the original spherical triangle. As a result, the edges of the four child spherical triangles and the parent spherical triangle do not coincide exactly. Thus, many researchers have chosen to take the midpoint of the great circle arc, which is the intersection point between the angle bisector of the two vertices of the great circle arc and the spherical surface. The spherical geodesic octree grid (SGOG) [42,43] performs a spherical surface subdivision of the great circle arc QTM and adopts an encoding structure consisting of “layer ID (hexadecimal) + octant ID (octal) + spherical position code (quaternary) + radial depth code (binary)”. The spherical position code adopts a fixed-direction code named the Goodchild code [44]. To avoid the conical surface generated by the latitude and radial lines and to make the cell shapes approximately consistent, the bottom edges of SGOG cells are designed as great circle arcs. However, the SGOG does not fully account for the geometric distortion such as area and volume distortion. The SGOG divides the radius equally in the radial subdivision without adjusting the spherical surface subdivision for each layer, resulting in significant geometric distortion of cells. The variable-length octree generates cells with approximate volumes by assigning different radial intervals to cells from the center to the spherical surface, resulting in significant distortions in cell shapes. The parallel method QTM [45] changes the bottom of each spherical triangle from a great circle arc into a latitude line and keeps the waists of the spherical triangles as great circle arcs. This change makes the area of the middle triangle more similar to that of the other three triangles, resulting in smaller grid distortion and higher data conversion efficiency. However, this change also leads to complex spatial relationships of cells, which generally exist in 3D spherical latitude–longitude grids, when the parallel method QTM is extended in the radial direction to divide the 3D sphere.

The existing ESSGs have the following limitations: (1) complex spatial relationships [25], (2) a large variety of cell shapes and sizes [24,46], and, in some cases, (3) designs that are specific to a certain field and difficult to apply in others [21,47]. Therefore, an ESSG suitable for comprehensively and systematically modeling, expressing, and analyzing the interior, surface, and exterior of the Earth is needed.

To reduce the geometric distortion and simplify the complex spatial relationships of cells in the existing ESSGs, a spherical geodesic degenerate octree grid (SGDOG) is proposed in this paper. This SGDOG extends the great circle arc QTM in the radial direction and adopts different levels of the great circle arc QTM at different radial depths. It adopts an encoding structure of “octant ID + radial depth code + spherical position codes”, along with dedicated encoding and decoding schemes. The experimental results show that the SGDOG has advantages of simple spatial relationships, convergent volume distortion, and real-time encoding and decoding.

2. Subdivision Method

2.1. Spherical Surface Subdivision

The SGDOG, as an extension of the great circle arc QTM in the radial direction, subdivides the spherical surface as follows:

1

Divide the sphere into eight octants using the longitude planes and the equatorial plane. Divide the southern hemisphere into octants 0 to 3 from longitude −180° to 180° with an interval of 90°, and the northern hemisphere into octants 4 to 7 in the same way, as shown in Figure 1. Each octant has a spherical triangular surface of equal area on the spherical surface, as shown in Figure 1.

2

Take the midpoints of the great circle arcs as midpoints and connect them with the original vertices using the great circle arcs to divide the parent spherical triangle into four child spherical triangles. To distinguish from the parallel method QTM, this method is called the great circle arc QTM. The edges of child spherical triangles are still on the edges of the parent triangle, as shown in Figure 2.

3

Repeat step 2 to generate the hierarchical levels of the great circle arc QTM.

2.2. 3D Spherical Subdivision

The SGDOG can be formed by combining different levels of the great circle arc QTM with radial subdivision as follows:

1

First-level subdivision. Bisect the radius lines and cut the octants with a spherical surface at half the original radius. The outer layers are quasi-triangular prisms, as shown in the red part of Figure 3a. The inner layers are quasi-triangular pyramids, that is, octants with half the radius of the original octants.

2

Second-level subdivision. Subdivide each outer quasi-triangular prism into 4 × 2 child quasi-triangular prisms where the great circle arc QTM partitions and radii bisect, as shown in the red part of Figure 3b. Subdivide each inner octant into an inner octant and a quasi-triangular prism following the method described in step 1.

3

Repeat step 2 until the SGDOG cell sizes meet the specified requirements.

The QTM is a quadtree, and the radial partitioning is a binary tree. If the whole spherical surface adopts the same level of QTM at a SGDOG subdivision level, the entire sphere will be an octree. However, different levels of QTM are adopted at different points on the radius, resulting in the grid at the center of the Earth having only five child grids (four quasi-triangular prisms and one quasi-triangular pyramid). Thus, the entire sphere is a degenerate octree.

3. Encoding and Decoding Schemes

Since the SGDOG bisects the radius during subdivision and its spherical layer can be calculated from the radial depth, it removes the layer ID from the SGOG encoding structure and adopts an encoding structure of “octant ID I + radial depth code J + spherical position code K” to reduce redundant codes.

The SGDOG employs Goodchild encoding for spherical position encoding, as shown in Figure 4, where 0, 1, 2, and 3 represent the child triangles at the center, top/bottom, left, and right, respectively.

The SGDOG employs the ETP conversion method and the distance comparison method to convert latitude–longitude coordinates into address codes. The principle involves comparing the distances between the point P(x, y) and the center points of four triangles and recording the address code of the triangle with the smallest distance until the distance is less than a specified threshold. For example, the first-level address code of point P is 3, as shown in Figure 5. The conversion from address code to latitude–longitude coordinates is the inverse process.

The variables in this section are defined as follows:

• Define the sphere radius as R.
• Define the level of SGDOG as nlevel.
• Define the level of the great circle arc QTM as n.
• Define the ETP projection coordinates as (x0, y0).
• Define the Cartesian coordinates of a point as $(\mathrm{x},\mathrm{y},\mathrm{z})$.
• Define the latitude–longitude coordinates of a point as $\left(\mathsf{\lambda },\mathrm{\phi },\mathrm{r}\right),$ where $\lambda$ is the longitude, $\phi$ is the latitude, and $r$ is the radial distance.
• Define the layer from the center to the spherical surface as radial depth J. For the innermost layer, n = 0 and J = 0.

3.1. Encoding Schemes

Given the sphere radius R and a point P$\left(\lambda ,\phi ,r\right),$ the steps to calculate the address code of P at nlevel are as follows:

1

Calculate the octant ID I from $(\mathsf{\lambda },\mathrm{\phi })$ using equation

$$\mathrm{I}={\displaystyle \frac{\mathsf{\lambda }+180°}{90°}}+4\times \mathrm{f}\mathrm{l}\mathrm{a}\mathrm{g},\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{g}=\left\{\begin{array}{c}0,\mathrm{\phi }\le 0\\ 1,\mathrm{\phi }>0\end{array}\right.,$$

(1)

where $\mathsf{\lambda }\in \left[-180°,\left.180°\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{\phi }\in \left[-90°,90°\right]\right.$.

2

Calculate the radial interval $\Delta \mathrm{r}$ of the cell using equation

$$\Delta \mathrm{r}={\displaystyle \frac{\mathrm{R}}{2^{\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}}}}.$$

(2)

3

Calculate the radial depth code J of P using equation

$$\mathrm{J}=\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}\left({\displaystyle \frac{\mathrm{r}}{\Delta \mathrm{r}}}\right),$$

(3)

where floor represents rounding down, as it does below.

4

Calculate the QTM level n of P using equations

$$\left\{\begin{array}{c}\mathrm{n}=\mathrm{ceil}\left({\log }_2\left(\mathrm{J}+1\right)\right),\mathrm{J}<2^{\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}}\\ \mathrm{n}={\log }_2\mathrm{J},\mathrm{J}=2^{\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}}\end{array}\right.,$$

(4)

$$\left\{\begin{array}{c}\mathrm{n}=\mathrm{ceil}\left({\log }_2\left(\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}\left({\displaystyle \frac{\mathrm{r}\times 2^{\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}}}{\mathrm{R}}}\right)+1\right)\right),\mathrm{r}<\mathrm{R}\\ \mathrm{n}=\mathrm{nlevel},\mathrm{r}=\mathrm{R}\end{array}\right.,$$

(5)

where ceil represents rounding up.

5

Convert the longitude and latitude $(\mathsf{\lambda },\mathrm{\phi })$ to ETP projection coordinates using Equation (6) [36], and obtain the spherical position code K (quaternary) by applying the distance comparison method.

$$\left\{\begin{array}{c}{\mathrm{x}}_0={\displaystyle \frac{2^{\mathrm{n}}}{\mathsf{\pi }}}\left[\mathrm{\phi }+2\mathsf{\lambda }\left(1-{\displaystyle \frac{2}{\mathsf{\pi }}}\mathrm{\phi }\right)\right]\\ {\mathrm{y}}_0={\displaystyle \frac{2^{\mathrm{n}}\sqrt{3}}{\mathsf{\pi }}}\mathrm{\phi }\end{array}\right.$$

(6)

6

Convert I, J, and K into binary and form a SGDOG code of 3 + nlevel + 2n bits according to

Table 1. The structure of SGDOG address code.

Octant ID I	Radial Depth Code J	Spherical Position Code K
3 bits	nlevel bits	2 × n bits

.

For example, given R = 6,371,000 m, nlevel = 10, and P (25.124°, 38.592°, 3,652,000 m), then I = 6 (decimal), J = 586 (decimal), and K = 0200010230 (quaternary). Convert I, J, and K to binary, resulting in I = 110, J = 1001001010, and K = 00100000000100101100. The SGDOG code for the grid containing P is 110100100101000100000000100101100.

3.2. Decoding Schemes

Decoding is the inverse process of encoding. Given R, nlevel, and the SGDOG address code of a grid, the steps to calculate the coordinates $\left(\mathsf{\lambda },\mathrm{\phi },\mathrm{r}\right)$ of the grid vertices are as follows:

1

Separate the octant ID I, radial depth code J, and spherical position code K from the 3 + nlevel + 2n bits address code. Convert I and J to decimal and K to quaternary.

2

Obtain the latitude and longitude ranges of octant I.

3

Calculate $\Delta \mathrm{r}$ using Equation (2) and then calculate the radii R1 and R2 of the inner and outer spherical surfaces of the cell using equation

$$\left\{\begin{array}{c}{\mathrm{R}}_1=\mathrm{J}\times \Delta \mathrm{r}\\ {\mathrm{R}}_2=\left(\mathrm{J}+1\right)\times \Delta \mathrm{r}\end{array}\right.$$

(7)

4

Sequentially read each digit of the spherical position code K in quaternary. Determine the position of the child triangle within the parent triangle and calculate the Cartesian coordinates of its three vertices at each level. The Cartesian coordinates of the three vertices on the inner spherical surface of the cell are obtained after reading the last digit. Convert the Cartesian coordinates into latitude–longitude coordinates using equation

$$\left\{\begin{array}{c}\mathrm{r}=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2}\\ \mathsf{\lambda }={\displaystyle \frac{180°}{\mathsf{\pi }}}\times {\tan }^{-1}{\displaystyle \frac{\mathrm{y}}{\mathrm{x}}}\\ \mathrm{\phi }=90°-{\displaystyle \frac{180°\times {\cos }^{-1}\frac{\mathrm{z}}{\mathrm{r}}}{\mathsf{\pi }}}\end{array}\right.$$

(8)

5

Each vertex on the outer spherical surface has the same latitude and longitude as its corresponding vertex on the inner spherical surface within a grid. If V1$\left(\mathsf{\lambda },\mathrm{\phi },\mathrm{R}1\right)$ is a vertex on the inner spherical surface, V2$\left(\mathsf{\lambda },\mathrm{\phi },\mathrm{R}2\right)$ is its corresponding vertex on the outer spherical surface. Convert $\left(\mathsf{\lambda },\mathrm{\phi },\mathrm{R}2\right)$ into Cartesian coordinates using Equation (9). The Cartesian coordinates and latitude–longitude coordinates of the three vertices on the outer spherical surface can be obtained. Note that a cell is the innermost cell and has only one vertex (0, 0, 0) on the inner spherical surface when J = 0. The vertices on the outer spherical surface of the innermost cell correspond to the three vertices on the inner spherical surface of the innermost quasi-triangular pyramid in the SGDOG.

$$\left\{\begin{array}{c}\mathrm{x}=\mathrm{r}\sin {\displaystyle \frac{\left(90°-\mathrm{\phi }\right)\mathsf{\pi }}{180°}}\cos \mathsf{\lambda }\\ \mathrm{y}=\mathrm{r}\sin {\displaystyle \frac{\left(90°-\mathrm{\phi }\right)\mathsf{\pi }}{180°}}\sin \mathsf{\lambda }\\ \mathrm{z}=\mathrm{r}\cos {\displaystyle \frac{\left(90°-\mathrm{\phi }\right)\mathsf{\pi }}{180°}}\end{array}\right.$$

(9)

For example, assume nlevel = 8 and the grid address code is 110/10100011/0111101101000010 in binary. According to step 1 and Table 1, separate I (bits 0th to 2th), J (bits 3th to 10th), and K (bits after 11th) from the address code, then I = 6 in decimal (110 in binary), J = 163 in decimal (10100011 in binary), and K = 13,231,002 in quaternary (0111101101000010 in binary). According to step 2, the latitude and longitude ranges are both 0° to 90° since I = 6. According to step 3, R1 = 4,056,535.15625 m and R2 = 4,081,421.875 m. According to step 4, the Cartesian coordinates of the three vertices on the inner spherical surface are (935,008 m, 2,049,860 m, 3,373,320 m), (960,902 m, 2,051,140 m, 3,365,260 m), and (935,221 m, 2,071,550 m, 3,359,990 m). According to step 5, the latitude–longitude coordinates of the three vertices on the inner spherical surface are $\left(65.480°,56.2611°,\mathrm{4,056,535.15625} \mathrm{m}\right),\left(64.8983°,56.0565°,\mathrm{4,056,535.15625} \mathrm{m}\right),$ and $\left(65.7028°,55.9235°,\mathrm{4,056,535.15625} \mathrm{m}\right)$; the Cartesian coordinates of the three vertices on the outer spherical surface are (940,745 m, 2,062,440 m, 3,394,020 m), (966,797 m, 2,063,730 m, 3,385,900 m), and (940,959 m, 2,084,260 m, 3,380,600 m); and the latitude–longitude coordinates of the three vertices on the outer spherical surface are $\left(65.4807°,56.2611°,\mathrm{4,081,421.875} \mathrm{m}\right),\left(64.8983°,56.0565°,\mathrm{4,081,421.875} \mathrm{m}\right),$ and $\left(65.7028°,55.9235°,\mathrm{4,081,421.875} \mathrm{m}\right)$.

4. Experiment

The experiment was conducted in the Visual Studio 2022 environment using the C++ programming language on a computer equipped with a 13th Gen Intel^® Core^TM i7-13700F 2.10 GHz processor, 16 GB of RAM, and an NVIDIA GeForce RTX 4060 graphics card.

4.1. Visualization

The SGDOG was visualized using Direct3D and C++ programming to clearly illustrate the partitioning method in the radial direction. Figure 6 retains only the bottom surface of the octant and shows that the radial depth J gradually increases from the inner layer to the outer layer. The vertical lines are the straight radial lines, and the horizontal lines are the curved great circle arcs. The partitioning structure of the bottom surface is a degenerate quadtree, where the grid at J = 0 has only three child grids. Figure 7 shows the 3D visualization of the entire sphere when nlevel = 1, 2, and 3. The innermost cells adjacent to the sphere center are quasi-triangular pyramids with spherical bottom surfaces. Other cells are quasi-triangular prisms with spherical inner and outer surfaces and three planar sides composed of great circle arcs and radial lines.

4.2. Volume Distortion

The QTM cells are spherical triangles. The area of a spherical triangle ABC can be calculated using equation

$$\mathrm{S}={\mathrm{R}}^2\left(\sum _{\mathrm{i}=1}^3{\propto }_{\mathrm{i}}-\mathsf{\pi }\right),$$

(10)

where R is the radius of the spherical surface containing the spherical triangle, and ${\propto }_{\mathrm{i}}$ is each interior angle of the spherical triangle.

Since each interior angle of the spherical triangle is a dihedral angle formed by the intersection of the planes in which the two great circle arcs lie, it can be calculated using equation

$${\propto }_{\mathrm{i}}={\cos }^{-1}{\displaystyle \frac{{\mathbf{X}}_{\mathbf{i}}\times {\mathbf{X}}_{\mathbf{i}-1}}{|{\mathbf{X}}_{\mathbf{i}}\times {\mathbf{X}}_{\mathbf{i}-1}|}}·{\displaystyle \frac{{\mathbf{X}}_{\mathbf{i}}\times {\mathbf{X}}_{\mathbf{i}+1}}{|{\mathbf{X}}_{\mathbf{i}}\times {\mathbf{X}}_{\mathbf{i}+1}|}},$$

(11)

where $\mathrm{i}\in \left[1,3\right],{\mathit{X}}_0={\mathit{X}}_3,\mathrm{a}\mathrm{n}\mathrm{d}{\mathit{X}}_4={\mathit{X}}_1$. ${\mathit{X}}_1{,\mathit{X}}_2{and\mathit{X}}_3$ are vectors of points A, B, and C, respectively.

Since the SGDOG extends the great circle arc QTM in the radial direction, the volumes of SGDOG cells can be calculated using equation

$$\mathrm{V}={\int }_{{\mathrm{R}}_1}^{{\mathrm{R}}_2}\mathrm{S}\mathrm{d}\mathrm{r},$$

(12)

$$\mathrm{V}={\displaystyle \frac{1}{3}}\left({\mathrm{R}}_2^3-{\mathrm{R}}_1^3\right)\left({\sum }_{\mathrm{i}=1}^3{\propto }_{\mathrm{i}}-\mathsf{\pi }\right),$$

(13)

where R1 and R2 are the radii of the inner and outer spherical surfaces of the cell, respectively, and R2 > R1.

The volume distortion of four ESSGs was compared: SGDOG, SDOG, H3-3D, GeoSOT-3D.

Given R = 6,371,000 m, the maximum volume V_max, minimum volume V_min, and the ratio V_max/V_min of the SGDOG, SDOG [25], H3-3D, and GeoSOT-3D [23] for levels 1 to 15 were calculated using Equation (12). H3-3D is generated by extending the H3 [33] in the radial direction using the same method as the SGDOG. As shown in

Table 2. Volume distortion.

nlevel	Radial Interval (m)	Vmin (m3)	Vmax (m3)	Vmax/Vmin
				SGDOG	SDOG	H3-3D	GeoSOT-3D
1	3,185,500.00	1.69251 × 1019	4.15802 × 1019	2.457	2.475	2.221	7
2	1,592,750.00	2.02396 × 1018	7.73473 × 1018	3.822	4.010	5.982	91.077
3	796,375.00	1.99239 × 1017	1.14169 × 1018	5.730	5.254	13.255	1847.22
4	398,187.50	2.20429 × 1016	1.53539 × 1017	6.965	6.043	26.246	6040.29
5	199,093.75	2.59035 × 1015	1.98548 × 1016	7.665	7.243	48.912	216,049
6	99,546.88	3.1389 × 1014	2.52262 × 1015	8.037	7.922	88.349	345,984
7	49,773.44	3.86297 × 1013	3.17854 × 1014	8.228	8.308	157.078	1.40 × 106
8	24,886.72	4.7912 × 1012	3.98889 × 1013	8.325	8.565	277.074	1.12 × 107
9	12,443.36	5.96566 × 1011	4.99592 × 1012	8.374	8.696	486.797	8.99 × 107
10	6221.68	7.44253 × 1010	6.25101 × 1011	8.399	8.770	853.622	1.01 × 109
11	3110.84	9.29409 × 109	7.81759 × 1010	8.411	8.821	1495.48	1.28 × 1010
12	1555.42	1.16119 × 109	9.77437 × 109	8.418	8.846	2618.37	3.46 × 1011
13	777.71	1.45114 × 108	1.22195 × 109	8.421	8.861	4585.35	3.45 × 1011
14	388.86	1.8137 × 107	1.52753 × 108	8.422	8.872	8033.09	2.76 × 1012
15	194.43	2.26699 × 106	1.90947 × 107	8.423	8.877	14,049.9	2.20 × 1013

and Figure 8, the V_max/V_min of the SGDOG increases slowly as the level increases and converges to 8.43, which is smaller than the convergence value 8.89 of the SDOG, while the V_max/V_min of H3-3D and GeoSOT-3D increases rapidly and diverges as the level increases.

4.3. Encoding and Decoding Efficiency

The encoding and decoding efficiency is defined as the number of encoding and decoding times per second.

Table 3. Encoding and decoding efficiency of SGDOG.

nlevel	Encoding Efficiency (times/s)	Decoding Efficiency (times/s)
1	147,859	180,342
2	74,360	110,776
3	48,061	83,951
4	35,062	68,442
5	27,614	57,478
6	22,882	50,835
7	19,299	45,348
8	16,841	40,504
9	14,882	37,323
10	13,361	34,541
11	12,068	31,775
12	11,018	30,075
13	10,103	27,909
14	9256	26,473
15	8680	24,895

presents the encoding and decoding efficiency for levels 1 to 15 of the SGDOG, tested with one million random latitude–longitude coordinates and one million random SGDOG address codes. The experimental results demonstrate that the encoding and decoding schemes can achieve real-time processing for spatial queries and data retrieval. The encoding and decoding efficiency decreases as the level increases, since the number of bits in address codes grows with the level. Encoding is less efficient than decoding since the ETP conversion method used in encoding requires determining the direction and involves calculations with irrational numbers and exponents.

5. Discussion

The SGDOG is compared with the SDOG, GeoSOT-3D, and H3-3D in terms of ESSG types, recursive subdivision methods, encoding schemes, and volume distortion, as shown in the

Table 4. Comparison of Four ESSGs.

	Types of ESSG	Recursive Subdivision Methods	Encoding Methods	Cell Shapes	Volume Distortion
SGDOG	3D spherical regular polyhedron grid	degenerate octree	hierarchical encoding	2 types	converge to 8.43
SDOG	3D spherical latitude–longitude grid	degenerate octree	Z-curve encoding	3 types	converge to 8.89
H3-3D	3D spherical regular polyhedron grid	degenerate fourteen-ary tree	unspecified	4 types	divergent
GeoSOT-3D	3D spherical latitude–longitude grid	complete octree	Z-curve encoding	3 types	divergent

.

5.1. Types of ESSG

The SGDOG and H3-3D are 3D spherical regular polyhedral grids, while the SDOG and GeoSOT-3D are 3D spherical latitude–longitude grids.

5.2. Recursive Subdivision Methods

Both the SGDOG and SDOG adopt a degenerate octree structure, while H3-3D adopts a degenerate fourteen-tree structure, and GeoSOT-3D adopts a complete octree structure.

5.3. Encoding Methods

The SGDOG adopts a hierarchical encoding scheme consisting of ”octant ID + radial depth code + and spherical position code”, while the SDOG and GeoSOT-3D adopt Z-curve encoding. H3-3D was designed by us for comparison by extending the H3 in the radial direction using the same method as the SGDOG. Thus, H3-3D does not have a corresponding encoding method. Since the encoding and decoding processes for I and J are inverse to each other, the encoding and decoding efficiency for I, J should be roughly the same. However, the experimental results show significant differences, with encoding efficiency consistently lower than decoding efficiency. This gap tends to widen as the level increases. It can be inferred that the primary reason for this efficiency discrepancy lies in the conversion process between the spherical position code K and the spatial coordinates. The encoding of K adopts the ETP conversion method, while the decoding process recursively identifies the midpoint of the great circle arc in the corresponding direction based on the Goodchild encoding. Although the ETP method is not the most efficient for calculating the QTM address code, it produces the most accurate results [48]. Since this paper prioritizes accuracy, the ETP method was adopted.

5.4. Cell Shapes

The SGDOG has two types of cell shapes: quasi-triangular pyramids next to the sphere center, and quasi-triangular prisms elsewhere. Both the SDOG and GeoSOT-3D have three types of cell shapes: quasi-triangular pyramids next to the sphere center, quasi-triangular prisms next to the poles in the outer layers, and quasi-quadrangular prisms elsewhere. H3-3D has four types of cell shapes: quasi-hexagonal pyramids and quasi-pentagonal pyramids next to the sphere center, and quasi-hexagonal prisms and quasi-pentagonal prisms elsewhere. Therefore, the SGDOG has the fewest types of cell shapes among these four ESSGs.

5.5. Volume Distortion

The Vmax/Vmin of the SGDOG increases slowly as the level increases and converges to 8.43, which is smaller than the convergence value 8.89 of the SDOG, while the Vmax/Vmin of H3-3D and GeoSOT-3D increases rapidly and diverges as the level increases. The SGDOG, SDOG, and H3-3D adopt different levels of spherical surface subdivision at different radius, while GeoSOT-3D adopts the same level of spherical surface subdivision for all the spherical surface. Thus, the Vmax/Vmin of GeoSOT increases most rapidly as the level increases. Although the maximum-to-minimum area ratio of H3 converges [33], when the radial differences of all grids are consistent, its seven-aperture characteristic causes the outer surface area of the grid at radial depth $\mathrm{J}=2^{\mathrm{k}}-1(\mathrm{k}\in \mathrm{Z},\mathrm{k}\ge 0)$ to be approximately 7 times that of the inner surface area of the grid at $\mathrm{J}=2^{\mathrm{k}}$. The spherical surface subdivision levels are the same from $\mathrm{J}=2^{\mathrm{k}}$ to $\mathrm{J}=2^{\mathrm{k}+1}-1$, and the outer surface area of grids at $\mathrm{J}=2^{\mathrm{k}+1}-1$ is about 4 times that of grids at $\mathrm{J}=2^{\mathrm{k}}$, while the outer surface area of grids at $\mathrm{J}=2^{\mathrm{k}}-1$ is about $7/4$ times that of grids at $\mathrm{J}=2^{\mathrm{k}+1}-1$. Consequently, the outer surface area of the innermost grid at J = 0 is approximately ${\left(7/4\right)}^{nlevel}$ times that of the outermost grid at $\mathrm{J}=2^{\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}}-1$. The higher the level, the faster the volume of the innermost grid increases, resulting in Vmax/Vmin divergence in H3-3D as the level increases. The SGDOG is based on the QTM, while the SDOG is based on the latitude–longitude spherical surface subdivision. The QTM has smaller area distortion than the latitude–longitude spherical surface subdivision. The smaller area distortion results in a corresponding smaller volume distortion. Therefore, the SGDOG has a smaller convergence value compared to the SDOG.

6. Conclusions and Future Work

To reduce geometric distortion and simplify the complex spatial relationships of cells in existing ESSGs, the SGDOG based on the great circle arc QTM is proposed and corresponding encoding and decoding schemes are designed in this paper. The SGDOG offers the following advantages:

1

Simple spatial relationships of cells. The spherical surface is subdivided by great circles, ensuring no curved surfaces are generated in the 3D subdivision. The SGDOG has only two types of cells shapes: quasi-triangular pyramids and quasi-triangular prisms. The relationships between cells and spatial rectangular coordinates are straightforward.

2

Explicit encoding structure. The SGDOG adopts an encoding structure of “octant ID + radial depth code + spherical position code”, enabling real-time conversion between latitude–longitude coordinates and cell address codes.

3

Convergent volume distortion. The volume distortion converges to 8.43, smaller than the convergence value 8.89 of the SDOG, and better than the divergent volume distortion of H3-3D and GeoSOT-3D.

Since the properties of spherical great circles and ellipsoid geodesics are similar, we suspect that the SGDOG can be applied to an ellipsoid. Further research is required to justify this conjecture and make necessary changes to the method. This paper primarily focuses on the conversion accuracy between the address code and the spatial coordinates, while the encoding and decoding efficiency can be improved in the future work. Moreover, we will utilize the SGDOG for spatial entity modeling, such as aircraft trajectory simulations and urban building modeling, which will facilitate the statistical analysis of spatial geometric properties.