Efficient cell navigation methods and applications of an aperture 4 hexagonal discrete global grid system

Abstract Hexagonal discrete global grid systems are multi-resolution frameworks for earth data and have attracted much attention in research towards the next generation of geographic information system. Each resolution of the framework divides the earth into a hexagonal grid of cells and cell codes are used instead of Cartesian coordinates to organize and query data. Aperture (the ratio of cells between resolutions) 4 systems have structural advantages compared with aperture 3 and 7, but haven’t been fully developed. In this manuscript, we examine the aperture 4 hexagon hierarchy on uniform tiles (HHUT) and modify the coding method. We propose two HHUT-based cell navigation methods, one for regular and the other for irregular areas. We demonstrate the utility and high efficiency of the proposed methods in two applications. Results show that the proposed method for an irregular area is on average 400 times faster than DGGRID, the conventional library. The proposed method for a regular area is on average seven times faster than the hexagon lattice quad tree method and approaches that of H3. The modified coding and novel cell navigation methods proposed here indicate high efficiency, superior data organization and broad application prospects.


Introduction
With the rapid development of earth observation and other sensor technologies, location-based data have grown exponentially. The data are multi-source, multi-scale and highly dimensional, leading to their characterization and aggregation becoming increasingly difficult (Wang et al. 2021). A Discrete Global Grid System (DGGS) recursively divides the entire earth's surface to form a multi-resolution discrete earth framework, which is conducive for establishing a new data organization model that emphasizes spatial location and is expected to support spatial data processing and analysis in Digital Earth frameworks (Sahr et al. 2003, Mahdavi-Amiri et al. 2015, Open Geospatial Consortium 2017. The cells of a DGGS are similar to numerous baskets distributed on the surface of the Earth. Compared with triangular and quadrilateral cells, hexagons have more ideal properties in that the shape is closer to a circle. This leads to higher efficiency while sampling circularly bandlimited signals (Sahr 2011). Hexagons also have the maximum number of directions, maximum angular resolution, minimum quantization error (Sahr 2003), and maximum number of aperture sequences. This maximum number of aperture sequences, defined as the ratio of cell areas between adjacent levels (i.e. apertures 3, 4, and 7), facilitates the flexible selection of resolution (Sahr 2019, Figure 1). These characteristics make the hexagonal cells more suitable for geospatial data organization and query than triangular and quadrilateral cells.
Considering geospatial data organization, either aperture 3, 4, or their combination is typically selected to obtain greater resolution and facilitate multi-scale data aggregation. This is because the area ratio between aperture-7 adjacent levels is large (7:1), making it difficult to select an appropriate cell resolution when aggregating data. Considering geospatial data query, aperture 7 cells are nearly congruent, making it facilitates efficient hierarchical operations. Compared with aperture 3, the orientation of aperture 4 does not change as the level changes, which is beneficial for searching cells between levels. Therefore, aperture 4 has certain structural advantages for data query compared with aperture 3 DGGS, and the cell resolution is smoother compared with aperture 7, making it a feasible solution for both organizing and querying multiscale geospatial data. However, the existing research focuses on the aperture 3 and aperture 7 DGGS, ignoring the excellent properties of the aperture 4 DGGS.
Following the DGGS terms in Open Geospatial Consortium (2017), we restate some definitions based on the requirements of this manuscript. Spatial referencing is achieved by addressing an identifier or a code to each DGGS cell. The coding method, that is, the method by which each cell is given a unique code that identifies its location and hierarchical relationship with other cells, affects the efficiency of spatial operations. Searching neighbors, namely finding neighborhood cells, is a basic operation that implements spatial analysis on the DGGS framework. Cell navigation methods, namely searching neighbors with specific methods for regular or irregular areas, are the underlying designs, directly determining the performance of geospatial data organization and query. Peterson (2011) designed the hierarchical coding method PYXIS for the icosahedral Snyder equal area aperture 3 hexagon. PYXIS is not open source and its cell navigation methods are unknown. Tong et al. (2013) proposed the aperture 4 hexagonal quaternary balanced structure (HQBS) and used cell codes to search neighbors. Experimental results showed that the HQBS neighbor search method was faster than that of PYXIS. Wang et al. (2020) proposed an aperture 4 hexagon lattice quad tree (HLQT) hierarchical coding method based on complex numbers, and the efficiency of searching neighbors was significantly higher than that of HQBS (Wang et al. 2021). However, PYXIS, HQBS and HLQT only focused on the simple operation of searching neighborhood cells for a single cell and did not involve methods of searching cells for areas. The open source project DGGRID (Sahr 2021) can generate multi-aperture hexagonal grids for data quantification and sampling after navigating cells for irregular areas. However, the cell navigation efficiency is relatively low, especially when the cell resolution is high. The commercial company Uber used an aperture 7 hexagonal DGGS to integrate taxi data and facilitate vehicle dispatch (Brodsky 2018). Its cell navigation method can be found in the open source project H3 (https://github.com/uber/h3). Zhou et al. (2020a) designed a cell navigation method based on HLQT and applied it to the query of land cover and point of interest data. The cell navigation efficiency was higher than that of DGGRID, and no comparison was made to H3. Therefore, most hexagonal DGGSs only consider searching neighborhood cells for a single cell, and there is a lack of research on cell navigation methods for areas. Moreover, the existing cell navigation methods of HLQT and DGGRID have the disadvantage of low efficiency.
DGGRID and H3 are state-of-the-art schemes in existing open source hexagonal DGGSs. DGGRID established coordinate-based codes on each face of the icosahedron. The coordinate-based code can search neighbors quickly on a single level, but it does not have multi-scale information, making it difficult to establish a hierarchical relationship. Li et al. (2021) used DGGRID to quantify Canadian heterogeneous digital elevation model data into a hexagonal DGGS. They showed that the coordinate-based code is beneficial for parallel processing, but it reduces efficiency when aggregating multiscale data. Bousquin (2021) applied DGGRID and H3 to the organization and processing of coastal water quality data. The efficiency of H3 was found to be higher, but DGGRID was found to provide more resolution options. Therefore, the hierarchy-based code is conducive for multi-scale data aggregation (such as H3 and HLQT); the coordinate-based code is conducive for searching neighbors and parallel algorithm design (such as DGGRID). A well-designed coding method should consider taking advantage of both. Zhou et al. (2021) proposed an aperture 4 hexagon hierarchy on uniform tiles (HHUT) and its hierarchical coding method. The present study is based on HHUT, we aim to achieve more efficient cell navigation methods for both multi-resolution data organization and query. The main work and contributions of this manuscript are as follows: 1. We further supplement and improve the HHUT hierarchy-based coding by adding coordinate-based coding and conversion method to allow existing hierarchy-based DGGS to leverage different coding methods (Section 2.1.2). 2. For a regular area, we propose two cell navigation methods (Section 2.2). This provides different cell navigation methods for different application scenarios.
3. For an irregular area, we propose a novel cell navigation method (Section 2.3). This provides method support for data organization and query in large scale DGGS. 4. In Sections 3 and 4, we compare the efficiency with DGGRID and HLQT to demonstrate that the cell navigation methods proposed here are more efficient under similar conditions of data organization ability. We compare the efficiency with H3 to present that close efficiency can be achieved while the data organization ability of our methods is stronger due to a smoother grid structure.

HHUT coding method and its improvement
HHUT classifies aperture 4 hexagonal cells on the surface of the icosahedron into two types: A and B: While refining to the next level, one parent cell of type A generates one type A and 12 type B children cells; and one type B cell generates only one type A children cell (process in Figure 2). The 12 cells formed by the initial refinement ( Figure 3(a)) are of type A, also known as 'tile', and are identified by the hexadecimal code digits 0-b ( Figure 4). We map the icosahedral HHUT cells of the first four levels to the earth according to Snyder equal area projection (Snyder 1992), as shown in Figure 3. The construction process of  2.1.1. Hierarchy-based coding HHUT uses a hexadecimal code digits set C ¼ f0, 1, 2, 3, 4, 5, 6, a, b, c, d, e, f g to encode cells. According to the refining rule, the children of type A cells directly add code digits of set C at the end of the parent code, and the children of type B cells directly add 0 at the end. The global coding can be designed using a 32-digit hexadecimal sequence (RTC), as shown in Figure 4.

Coordinate-based coding and conversion method
The hierarchy-based code has obvious advantages in data aggregation, but the coordinate-based code is conducive for searching neighbors and parallel algorithm design (Li et al. 2021, Sahr 2021. Therefore, the coordinate-based code is added as a supplementary coding for HHUT to improve its coding method. A two-dimensional coordinate system is established on each rhombic surface of the icosahedron, and the cell can be uniquely identified by the coordinate-based code ði, jÞ ( Figure 5). These two types of coding methods have their own advantages; moreover, they complement each other in different scenarios through fast conversion, which also meets the functional requirement for code interoperability operation (Open Geospatial Consortium 2017). As will be discussed below, the coordinate-based code serves a matrix search-like cell navigation method (Section 2.2.1), and the hierarchy-based code serves a ring search-like cell navigation method (Section 2.2.2).
2.1.2.1. Converting hierarchy-based codes to coordinate-based codes. The tile integer coordinate Tðu, vÞ is the link between the hierarchy-based code RT Á Á Á C 1 C nÀ1 C n and the coordinate-based code qði, jÞ (q for the icosahedral rhombus, Figure 6). Hence, we first calculate Tðu, vÞ corresponding to RT Á Á Á C 1 C nÀ1 C n : Consider tile 4 in Figure 4 and its associated rhombuses as an example. According to the principle of complex number coding (Ben et al. 2018), hierarchy-based code digits have a natural relationship with the tile integer coordinates, as shown in formula (1) ( Figure 6).
where U u ðu ¼ 1, 2 Á Á Á , nÞ is the unit tile integer coordinate ðu 0 , v 0 Þ corresponding to the hierarchy-based code digit C c ðc ¼ 1, 2 Á Á Á , nÞ, as shown in Table 1. After Tðu, vÞ is calculated, it is converted into coordinate-based code. For any Tðu, vÞ, the corresponding qði, jÞ can be obtained using the following steps (m ¼ 2 n in the following expression) ( Figure 6): where T leftdown represents the tile number at the bottom left of tile T; 3. When u<0, v<0 (corresponding to q where T left represents the tile number at the left of tile T; 4. When u<0, v!0 (corresponding to T leftup represents the tile number at the upper left of tile T: 2.1.2.2. Converting coordinate-based codes to hierarchy-based codes. The conversion of coordinate-based codes to hierarchy-based codes is the inverse process of the above conversion, and the key is to calculate Tðu, vÞ corresponding to qði, jÞ: After obtaining Tðu, vÞ, we calculated the hierarchy-based code according to code addition operation (Zhou et al. 2021). To the best of our knowledge, the code conversion methods of Section 2.1. 2 have not yet been provided in the existing literature.

Cell navigation for a regular area
We design two cell navigation methods for a regular area. The regular area, namely a regular diamond area and a regular hexagon/pentagon area, corresponding to matrix search-like cell navigation and ring search-like cell navigation respectively.

Matrix search-like cell navigation method
The arrangement of coordinate-based codes is similar enough to a matrix to use a matrix search-like structure on the surface of an icosahedron. If a target geographic object is on one rhombic surface a regular diamond searching area is searched row by row and column by column, as shown in the purple cells in Figure 7(a). For a certain target geographic object (shown in translucent white), the searching range of the matrix structure (i.e. the coordinate-based codes of A and D), is calculated by converting the coordinates of the geographic object boundary into coordinate-based codes and sorting them individually. Assume the coordinate-based codes after sorting are Aði A , j A Þ and Dði D , j D Þ respectively; then, the searching range of the matrix is Að0, 0Þ, Bð0, JÞ, CðI, 0Þ, DðI, JÞ, where I ¼ i D Ài A , J ¼ j D Àj A , and the actual coordinate-based code needs to add an offset ði A , j A Þ: If the target geographic object extends across multiple rhombus surfaces, the matrix must be calculated separately for each rhombus. In such instances, a  Table 1. Mathematical relationship between C c and ðu 0 , v 0 Þ: distributed cell navigation method is adopted on each rhombus surface (Figure 7(b)). The matrix search-like cell navigation only needs to increase or decrease the integers of coordinate-based codes, allowing extremely high calculation efficiency, which is conducive to the sequential access of cells.

Ring search-like cell navigation method
As shown in Figure 4, the hierarchy-based coding has a characteristic of central symmetry. According to this property, a ring search-like cell navigation method is designed around a center point. Given a starting position as the center point and searching outward in a rotating manner to finally form a regular hexagonal/pentagonal area (Figure 8).
The ring search-like cell navigation method is divided into two parts: on the same ring and between adjacent rings. Starting with a given cell, the 0th ring only contains that cell. On the l th ring (l ¼ 1, 2 Á Á Á) there are l cells to consider for each of the six directions. We navigate between rings by searching neighbors in direction 6 ( Figure  8). In addition to the same ring and adjacent rings, cell navigation may either be within a rhombus face (Figure 8(a)) or cross rhombus faces (Figure 8(b)). The operation performs four steps: 1. The starting code for the 0th ring and the total number of rings k to expand outward are calculated. The number of rings k needs to be jointly determined according to the searching radius r and the cell resolution. Formula (2) shows that r and k have the following relationship, where d n is the distance between adjacent cell centers. (2) 2. The first cell code of the next ring is obtained from the initial code of the previous ring through searching neighbors in direction 6. 3. The cells of ring l are individually searched. Each ring needs to search neighbors in six different directions, such as 6 ! 1, 1 ! 2, 2 ! 3, 3 ! 4, 4 ! 5, 5 ! 6 in ring 1. After the ring navigation is complete, step (2) is repeated. 4.
Step (3) is repeated while l<k: Cell navigation requires special handling when the process involves crossing rhombus faces. As shown in Figure 8(b), the ring structure intersects rhombuses 1, 2, and 6 (the positional relationship can be obtained from Figure 4), where direction 3 is searched only once on each ring. This is because cell / and u are located between the rhombus faces, and a similar situation occurs between the upper triangles of rhombuses 1-5 and the lower triangles of rhombuses 6-10 (see in Figure 4).
We propose a multiplication-based rotation method when searching neighbors of cross-rhombus cells. Multiplication operation corresponds to rotation (Vince 2006), see supplementary materials for details. The cross-rhombus situation in Figure 8(b) is expanded on the plane as shown in Figure 9 (level 2, derived from Figure 4). Due to the uniform tile structure of HHUT, the cross-face cell and its corresponding cell on another triangular face are still in the same parent cell, such as 210 and 220, 201 and 202 in Figure 9. Therefore, we first rotate (perform multiplication operation) the crossface cell to another triangular face to obtain a virtual cell code (for example, / rotates 60 counterclockwise to obtain a virtual cell / 0 ), and then search neighbors within face by the virtual cell (for example, cell 220 search neighbors within face to get cell 20b and 00b). Similar processing is carried out for other cross-rhombus situations, such as the rhombuses 6-10, are rotated 60 clockwise.

Cell navigation for an irregular area
Notably, an irregular area has unequal side lengths and angles, which is surrounded by a series of boundary lines, and the boundary lines are connected to each other by a number of boundary points (Figure 10). The above cell navigation methods are used to search  for cells in a regular area. In practical applications we need navigate cells for an irregular area, such as boundaries for countries or continents, making consideration of all DGGS cells time consuming. Efficiency can be gained by reducing the set of cells considered as much as possible. To do this, the cells of the irregular area are divided into two types: cells that intersect the area and those that are contained within the area. We propose first determining the intersection cells and then searching the interior cells of the area. The cells are navigated according to the following steps: Figure  10, and sort them to determine the matrix range, ABCD in Figure 10. Intersection cells are determined by converting boundary points from their geographic coordinates ðB 1 , L 1 Þ, ðB 2 , L 2 Þ Á Á Á ðB m , L m Þ to their coordinate-based codes ði 1 , j 1 Þ, ði 2 , j 2 Þ Á Á Á ði m , j m Þ: 2. Extract the intersection cells for lines between the boundary points, blue cells in Figure 10. We consider the adjacent boundary points to be connected by a straight line and utilize the vector line generation algorithm of triangular grids (Du et al. 2018) to identify all cells between the boundary points. We then record the coordinatebased codes. Figure 10. The matrix searchlike cell navigation method is used to search cells in the matrix, and the interior cells are navigated according to the intersection cells. The specific process is as follows: if the current cell cði, jÞ intersects and the following cell cði, j þ 1Þ does not intersect, examine the positional relationship between cði, j þ 1Þ and the irregular area, and if it is within the area, then cði, j þ 1Þ is an interior cell, then skip cði, j þ 1Þ in the next navigation; if cði, jÞ does not intersect and the cell cði, jÀ1Þ is in the area, then cði, jÞ is an interior cell. From the above operation, it can be seen that the judgment of whether a cell is within the irregular area is only possible at the intersection of each row, which greatly reduces the amount of floating number calculations. The entire process of cell navigation for an irregular area is shown in Figure 11.

Experimental design
To demonstrate the advantages of the proposed methods, we test the cell navigation methods for regular and irregular areas respectively. Each of these tests includes two parts: quantitative comparison test and application case experiment. In this study, the matrix search-like cell navigation method mainly serves as a basis for navigation for irregular areas, hence, we do not set up a quantitative comparative test for the matrix search-like cell navigation method. The experimental environment is set as follows: Each program was compiled for the release version and operated on the same desktop (Core i7-8700@3.20 GHz six-core CPU, 8 G RAM WDC WDS480G2G0A-00JH30 480 G SSD; Windows 7 X64 Ultimate SP1; development software: Microsoft Visual Cþþ Enterprise 2017).

Quantitative comparison test
In the existing aperture 4 hexagonal DGGSs, the efficiency of HLQT is higher than that of PYXIS and HQBS (Wang et al. 2020). Therefore, this study compares the efficiency with the HLQT-based cell navigation method (Zhou et al. 2020a). The cell navigation method used by H3 is also useful, but direct comparisons cannot be made because it uses aperture 7. To obtain some comparability we chose HHUT levels with cell resolutions as close as possible to H3. For instance, the cell's average area of level 8 in H3 is 0.73732 km 2 , which is closest to level 13 in HHUT. The data points coded is Beijing, China, with the query starting position set to 40:1528 N, 116:3903 E: To avoid cross-rhombus processing and reduce the grid deformation, we place China in a single triangle of the icosahedron with the orientation parameters of V 0 ð55:8635 N, 43:6498 WÞ, a ¼ À129:86 (Zhou et al. 2020b). The grid level is selected from the 13th to the 20th, and the number of rings at each level is specified as k ¼ 100: The number of cells to be navigated is 3 Â k Â ðk þ 1Þ þ 1 ¼ 90301, and tests the total time for the entire process.

Application case experiment
The ring search-like cell navigation method can be applied to query codes within a certain range around a specified spatial location. Typical examples include querying data such as taxis and restaurants within 3 km of the user. Here, we consider querying the locations of shared bicycles as an example. We quantify the queried data into the HHUT cells and use the ring search-like cell navigation method to query the codes of the relevant area and the number of bicycles in the cell. The bike-sharing data from Dongcheng district in Beijing was provided by the Beijing Engineering Laboratory of Beidou Navigation and Positioning Technology. The 20th level is selected to quantify the sharing bicycle data, and the center distance of adjacent cells is 7.36 m, which meets the actual application requirements. We consider the user's location coordinates to be 39:9020 N, 116:4140 E, and the query range to be k ¼ 100; according to formula (2), the query radius is approximately 736 m. We query the bicycle positions from 7 am to 8 am and 11 am to 12 am.

Quantitative comparison test
The irregular area for the experiment is the continental United States (https://gdal.org/ drivers/vector/tiger.html), which has a large area and a long coastline. In DGGRID, Sahr (2021) implemented the grids generation algorithm for local irregular areas. Note that the grid generation algorithm includes two steps: the first is to navigate cells for the area and the second is to calculate the boundary, vertex coordinates of cells. This manuscript compares the efficiency with DGGRID's cell navigation method, namely the first step of DGGRID's aperture 4 hexagonal local grids generation algorithm, and does not include the handling of file input and output as in the second step. The selected grid level is from 6 to 12 (computation time in seconds).

Application case experiment
Using DGGS to organize the existing planar data can establish a spherical data model of a unified spatial datum. In this study, the earth observation data are quantified into the cells for establishing the global unified organization of the hexagonal DGGS. We aim to verify the correctness of the cell navigation method for irregular areas in multiscale spatial data organization. The experimental data were 2017 population data obtained from the LandScan Program (https://landscan.ornl.gov/) released by the Oak Ridge National Laboratory in the United States, with a resolution of 1km Â 1km: The selected grid level is from 12 to 13 and the irregular area is the continental United States. The basic quantification process is as follows: First, map the original data from its geographic coordinates to the cell codes, and bind the corresponding attribute value to the codes simultaneously. Second, navigate the codes, and interpolate the attribute values (take the average) in a certain neighboring range, which corresponds to k ¼ 2 in the ring search-like cell navigation method.  Table 2, and the comparison histogram is shown in Figure 12. The time unit is milliseconds (ms), and the unit of query range is km: The relation between query range and k can be found in formula (2). The results suggest that the ring search-like cell navigation method based on HHUT is more efficient than that of HLQT from levels 13-20. The average efficiency is approximately seven times that of the HLQT-based method (Table 2), and as the level increases, HLQT-based method consumes more time than the proposed method ( Figure 12). There are two main reasons for these results. (1) Inside the tile, the neighbors searching efficiency of HLQT is lower than that of HHUT, because the neighbors searching method of HLQT generates more carry digits than that of HHUT (Wang et al. 2020), and requires more calculation source. (2) HLQT contains 12 vertex tiles and 20 face tiles, which leads to more frequent and complex cross-tile processing (Zhou et al. 2020a); HHUT has a uniform tile structure (Figure 3(a)), which can perform cross-tile processing uniformly. The average cell navigation efficiency of H3 is approximately twice that of the proposed method. This is because the aperture 7 grids have a nearly congruent structure.

Application case experiment
The experimental results are visually rendered using Mapbox, as shown in Figure 13. The greater the number of bicycles in a cell, the darker the obtained color.
The overall query results in Figure 13 form a regular hexagonal shape consistent with the characteristics of the ring search-like cell navigation method (Figure 8). It can be judged by the color of cells, the number of bikes during rush hour periods ( Figure   Figure 12. Efficiency test results for comparison. 13(a)) is significantly higher than that in the noon (Figure 13(b)), which is closely related to human social activities.

Quantitative comparison test
We first convert cells that intersect points on the area boundary and then extract cells for lines connecting boundary points. Meanwhile, the coordinate-based codes are sorted, and the matrix structure is obtained (Figure 14(a), level 6). During the cell navigation process, we search the interior cells. The visualization results are shown in Figure 14(b,c), in which the interior and boundary cells can be distinguished. From the results of level 8, the zoomed-in result of the local area near the eastern coast is shown in Figure 14(c). The results show that the searched boundary cells fit the boundary line well, thus verifying the accuracy and feasibility of our method. The efficiency test results of DGGRID and our method are shown in Table 3 and Figure 15.
This study successfully utilizes the matrix search-like cell navigation method ( Figure  14(a)) to assign boundary and interior cells (Figure 14(b,c)). When compared with DGGRID ( Figure 15), the proposed method is significantly faster at navigating irregular area. The average efficiency of the proposed method is approximately 400 times that of DGGRID from levels 6 to 12 (Table 3), and the advantages of the proposed method are more significant as the level increases. The specific reasons are as follows.
First, DGGRID clips the irregular area into polygons, and then determines whether the cell intersects with the polygons individually. Intersection judgment requires a large amount of floating calculations, which is time-consuming when perform for a large number of cells. Therefore, the time required by DGGRID increases with the increase in grid level and is positively correlated with the number of cells. Thus, the time-consumption ratio of adjacent levels is approximately 4:1 (Table 3).
Second, in the proposed method, the positional relationship between the cell and the irregular area is only determined at the intersection of the boundary, and the cells contained by the area are completely searched by matrix search-like cell navigation method. Therefore, the proposed method is efficient and not sensitive to the number of cells. Thus, the time-consumption ratio between adjacent levels is approximately 2:1.

Application case experiment
We apply the cell navigation method (Section 2.3) to organize population data into the hexagonal DGGS. Level 11 national results are shown in Figure 16(a), with magnified results for New York City. The same magnified area near New York city is shown for level 12 (Figure 16(b)) and level 13 (Figure 16(c)). The area of level 13 (average cell area: 0.76 km 2 ) are closest to that of the original data (1.0 km 2 ).
When integrating multi-source datasets for analysis in a large area, datasets based on planar projections are often inconvenient for integrated analysis due to multiple  projection datums. This case restores the flat population data to what it should be: directly related to the real area on the earth, which is conducive to the joint processing and analysis of large areas due to unified spatial datum. From the results, our cell navigation methods of aperture 4 DGGS can provide a more resolution data organization scheme, with more levels, relative to aperture 7 DGGS.

Discussions and conclusions
DGGS is expected to become the underlying data frame of a new generation of geographic information systems. The efficient coding and cell navigation methods are at the foundation of any DGGS. This work emphasizes the value of the aperture 4 hexagonal DGGS and proposes a combination of hierarchy and coordinate-based coding methods. Further, based on these two coding methods, we propose two cell navigation methods suitable for regular and irregular areas, and we demonstrate their efficiency using different application scenarios. The proposed cell navigation methods here have the following advantages: (1) they make up for the inefficiency of DGGRID and HLQT and (2) avoid the limitations of H3 for multiscale data organization. H3 showed good stability, with little change in computational time with level increases. These advantages support the adoption of H3 by Uber. However, H3 is an aperture 7 DGGS, which causes great variation in the inter-level cell resolution. Thus, although the efficiency is only one-half of the aperture 7 H3, HHUT has an inherent advantage for data organization considering the grid structure. The DGGS, especially the hexagonal DGGS, is not only useful for spatial indexing but also for integrating, fusing, and processing multi-source and multi-scale spatial data. Therefore, we believe that because the HHUT scheme is more suitable for organizing, managing, and querying geospatial data, its low spatial indexing efficiency loss can be tolerated.
The proposed cell navigation method for irregular areas can establish multi-scale organization of earth observation data on DGGS. The smoother aperture of HHUT facilitates the organization of multi-scale datasets and multi-precision quantization. Data sets within different resolutions have their own application value. High-resolution data are used to mine reliable information, and low-resolution data facilitate fast transmission and can be converted between scales losslessly through Fourier transforms (Mahdavi Amiri 2019). As per our proposed methods, any other location-related earth observation data can also be quantified and organized in DGGS, which is beneficial to the process of multivariate heterogeneous data.
Therefore, DGGS provides a new idea for the organization and query of spatial data in the field of geographic information system; essentially, the data are unified on the surface of the earth, rather than under the framework of local projection. Moreover, although this work focuses on hexagonal DGGS, the coding and cell navigation methods are also applicable to triangle and quadrilateral DGGS, because triangles and hexagons have a dual relationship, and two triangles can be combined into one quadrilateral.
Future research should consider methods to quantify raster and vector data more efficiently and accurately into DGGS, thus building on the preliminary exploration carried out in this study. The establishment of a spatial database associated with DGGS to facilitate real-time query and analysis of data is necessary.
Data and codes availability statement