Discrete Global Grid Systems (DGGSs) utilize a hierarchical partition of the Earth's surface into a set of discrete cells, providing a globally consistent structure for cell-based indexing, fusion, and multi-resolution processing of geospatial data. Current approaches to mapping remotely sensed imagery to DGGSs predominantly rely on resampling orthorectified products. This introduces procedural redundancy and information loss, resulting in slower data ingestion and reduced accuracy. In this study, we propose a direct orthorectification method from level-1 imagery to DGGSs based on a rational function model. We also enhance the polyhedral equal-area projection to address cross-face imaging cases, and develop a key technology for generating DGGS-orthorectified products directly from source data. Using a rhombic triacontahedron 'slice-and-dice' equal-area hexagonal grid system as a representative DGGS implementation, we conduct experiments using Gaofen-1 and Gaofen-2 satellite imagery. Our results demonstrate that, under the nearest-neighbor interpolation method, the proposed approach reduces mean squared error by 52.98-59.25% and improves the structural similarity index by 33.53-101.47% relative to level-1 reference data obtained with conventional methods, thereby significantly enhancing the radiometric accuracy of gridded products. This approach establishes a novel paradigm for organizing analysis-ready data within DGGSs.

Rhombic triacontahedron-based hexagonal discrete global grid systems (RTHDGGS) exhibit desirable geometric properties; however, their practical application is constrained by expensive cross-face computations and incongruent hierarchical subdivisions. To overcome these limitations, we propose a combined structure that merges three adjacent rhombic faces sharing one vertex and defines a triaxial integer coordinate system for unified spatial encoding. Building on this structure, we developed a multiscale model based on the aperture-4 hexagonal subdivision, including topological mapping rules for subdivision consistency and a coordinate conversion mechanism for efficient cross-scale aggregation and decomposition. Experiments show that the proposed method retains the geometric advantages of the icosahedron, improves adaptability to regions of interest, increases regional grid generation efficiency by approximately 2.3 times, and reduces hexagonal data aggregation error by 40% relative to previous methods. The proposed framework supports efficient and robust geospatial computing for large-scale Earth system applications.

The discrete global grid system (DGGS), a multi-resolution hierarchical structure constructed by recursively dividing the Earth's surface, has promising applications in wide-area environmental modeling and geospatial computing. However, existing DGGS research faces several limitations. First, most DGGS implementations are restricted to a single grid-cell shape, such as triangular, rhombic, or hexagonal, which limits the flexibility to adapt grid structures to diverse application requirements. Second, current approaches predominantly focus on grid centers, neglecting the importance of vertices and edges (grid centers, vertices, and edges are collectively referred to as multi-structural elements). To address these challenges, this paper proposes a dual-structure approach to construct dual grid systems with multi-structural elements (DGSME) based on three regular polyhedra: the regular octahedron, icosahedron, and rhombic triacontahedron. DGSME supports hierarchical subdivision while preserving consistent orientation, thereby improving scalability and computational efficiency. The proposed framework enables the unique representation of multi-structural elements. As demonstrated in the case study, incorporating vertices and edges into the grid representation facilitates more detailed environmental modeling, yielding route planning outcomes with lower costs and reduced error compared to traditional single grid-centered models.

The discrete global grid system (DGGS) is a digital Earth reference framework that supports the fusion, analysis, and processing of heterogeneous data from multiple sources. Hexagonal DGGS is suitable for multi-scale data representation, with commercial and theoretical applications. The establishment of a hierarchical structure and design of encoding operations for hexagonal DGGS are vital for improving application efficiency; however, challenges remain in improving the efficiency of coding operations due to a lack of theoretical support for algorithm design. Therefore, this study describes the spatial cognitive equivalence of the hexagonal hierarchical division as a hybrid partition of triangles and hexagons in a T(6.3.6.3) grid, establishes a “superimposed” partition model of the aperture-3 hexagonal grid, and proposes a coding mathematical model of finite boundary structure, namely hexagonal hierarchy in a finite boundary (HHFB), which is extended to the rhombic triacontahedron and spherical surfaces. The comparative experimental results show that the proposed scheme has a smooth transition at the tile boundary and avoids complex fractal overlay hierarchy. The HHFB scheme achieves 10-fold improvement in encoding addition efficiency compared to PYXIS, with neighbor query speeds reaching sixfold and 12-fold acceleration at odd and even levels, respectively.

The icosahedron is currently the mainstream polygon in research and application of discrete global grid systems (DGGS). However, compared to the rhombic triacontahedron (RT), the icosahedron has disadvantages, such as lower sphere-fitting accuracy, greater projection distortion, and difficulty in incorporating the matrix structure for geospatial data storage. More importantly, the special positional relationship between the rhombic triacontahedron and the Earth enables it to effectively support event simulations related to geographical locations. To this end, bidirectional mapping of the hexagonal grid between the RT and icosahedron was proposed, which can efficiently integrate the existing datasets and algorithms of icosahedral DGGS into RT DGGS, thereby achieving seamless conversion between heterogeneous grid systems. We established geometric and topological correlations between the RT and icosahedron, abstracted the spatial algebraic structures of hexagonal grids on the two different polygons, and constructed mapping relationships between them. Finally, conversion between heterogeneous grid indices was achieved using dual quaternions. Experiments revealed that the proposed method was 3.9150 and 2.8151 times more efficient at grid conversion from RT to icosahedron and from icosahedron to RT, respectively, than was a method using latitude/longitude coordinates as a medium.

Discrete Global Grid Systems (DGGSs) employ uniform cells for hierarchical Earth surface representation. However, prevalent software systems stemming from DGGS only focus on grid centers and lack the modeling of vertices and edges, which hindering efficient geospatial data computation and analysis. Consequently, we propose a novel mathematical model for hexagonal centers, vertices and edges of an icosahedral DGGS and provide corresponding hierarchical indexing methods and conversion rules with respect to integer coordinates. The proposed method was incorporated into the open-source software DGGRID, thereby enhancing its capabilities by integrating data fusion into the computation framework and expanding its application scope. The results show that the introduction of multi-structural elements could lead to higher-precision modeling of raster and vector data.

Discrete Global Grid Systems (DGGSs) are an emerging Earth reference model that support the integration and analysis of remote sensing (RS) data. Grid modeling, sampling, quantization, and storage are the key points and difficulties of DGGS. The Icosahedral Snyder Equal-area Aperture-4 Hexagonal DGGS was introduced as the basic framework to improve the geospace sampling efficiency. We presented an approach for precise hexagonal pixel modeling and an easy-sharing storage scheme compatible with open-standard formats for RS images based on the DGGS. Firstly, the proposed interpolation is computed by overlaps between quadrilateral and hexagonal pixels. The hexagonal grids were then mapped onto the icosahedral surface, and a strict correspondence between hexagonal and rectangular pixels is established. The open-standard formats were used to accurately store the hexagonal attribute values and metadata. Finally, a multiscale hexagonal aggregation algorithm based on the dataset was designed. Experiments showed that the proposed modeling methodology had a higher accuracy and smaller errors. The hexagonal data can be stored as regular rectangles with a fixed pattern in any standard format. This storage scheme was more conducive to data processing and sharing compared to SMOS, which was expected to promote hexagonal DGGS in RS data organization, processing and sharing.

One of the basic scientific problems concerning geographic information science is how to rapidly organize, query, and compute spatiotemporal big data. The spatiotemporal discrete global grid system (DGGS) provides a homogenized discrete structure for processing multiscale and multitype spatiotemporal data. To date, most research in spatiotemporal DGGS has focused on spatial discretization while neglecting temporal discretization. Here, we propose a general modeling scheme for spatiotemporal DGGS with emphasis on encoding and operating multiscale time grids. We subdivide continuous time into multiscale temporal grids, which are then encoded as integers. Moreover, we designed integer code operations, including hierarchical traversal, neighborhood finding, and temporal relationship calculations. Compared to the multiscale time segment integer coding (MTSIC) approach, the proposed method resulted in 22% higher encoding efficiency, 10.92 times faster decoding, 2.81 times better parent code finding efficiency, 41% improved efficiency, 100% accuracy in finding children codes (compared to less than 100% with MTSIC), and a 62% enhancement in temporal relationship calculation efficiency. The application of querying spatiotemporal trajectory data validates the feasibility and practicality of substituting conventional string-based time and floating-point location coordinates with spatiotemporal integer codes to query data. The time encoding and operation methods proposed here indicate high efficiency, superior accuracy, and broad application prospects.

Discrete Global Grid System (DGGS) is a hierarchical structure with seamless global coverage, supporting processing and analysis of heterogeneous geospatial data. Compared with the most commonly used icosahedron, the rhombic triacontahedron (RT) approximates Earth better and can improve the accuracy of data modeling and expression. However, current RT DGGS studies have adopted a single-resolution integer coordinate scheme within a single surface that cannot achieve multilevel grid analysis. To this end, three adjacent surfaces are combined and a three-axis integer coordinate system is established to reduce the number of times crossing the surface. Then, the principle for partitioning aperture 4 hexagonal grids and a quadtree hierarchical encoding scheme are proposed. Further, this study establishes a fast transformation between the code and three-axis coordinates, implements hierarchical and neighbor code operations, and designs a transformation between the code and geographic coordinates. Experimental results indicate that, compared with the icosahedral hexagonal DGGS, the proposed scheme has a significant efficiency advantage in the transformation between the code and geographic coordinates, while the neighbor code operation efficiency can reach 40.97 times that of HLQT and 4.35 times that of HHUT. The proposed scheme provides a more suitable framework for organizing and analyzing global geospatial data.

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.

全球离散格网系统(Discrete Global Grid System,DGGS)是数字化的地球参考框架,在多源、多尺度地球空间数据集成分析方面优势明显.本文选择菱形三十面体六边形全球离散格网系统,提高格网与地球的整体拟合精度和空间采样率;建立遥感图像六边形像素数学模型,提出兼容开放标准格式的数据存储方案.①根据地理位置将遥感图像格网化,完成遥感图像六边形DGGS建模;其次,建立六边形单元与矩形像素严密对应关系,等效保留六边形单元的邻域信息;②采用GeoTIFF开放标准格式精确存储六边形属性值以及投影、变换参数;③设计依托六边形DGGS格网标准数据集为基础的多尺度六边形DGGS生成算法.实验结果表明:本文方案不仅能保证六边形像素遥感图像数据与标准文件格式兼容,而且能保证矩形像素与六边形单元逐一对应,较好地保留了六边形单元数据的图像信息和空间分布特征,相较于欧空局SMOS数据组织方案更具优势.本文方案打破了六边形单元与矩形像素遥感图像的数据组织壁垒,使用常见GIS/RS软件即可读取六边形像素的遥感图像,并可通过对矩形像素的操作等效实现对六边形单元的处理,有望推动六边形DGGS在遥感数据组织、处理、共享等方面的应用.

Objectives: Discrete global grid systems are the preferred data models supporting multisource geospatial information fusion. Hexagonal grids have become more popular in many applications due to their geometric characteristics within uniform adjacent. Methods: We design a uniform tiles hierarchy on the surface of the icosahedron according to the characteristics of the aperture-4 hexagonal discrete global grids, using complex numbers to build a unified coding and operation model. We also design algorithms including interoperating between geographic coordinates and codes, querying neighborhood codes. Results: The experimental results show that interoperation between geographic coordinates and codes efficiency of the proposed algorithm is approximately 2.74 and 1.73 times that of the traditional algorithm respectively,and that neighborhood codes query efficiency of the proposed algorithm is approximately 7.46 times that of the traditional algorithm. As the grid level rises, the advantages of the proposed algorithm become more obvious.Conclusions: The results of this paper are expected to provide theoretical and technical supports for the unified organization, management, processing and analysis of multi-source earth observation data.

Discrete Global Grid Systems (DGGS) provide a multi-resolution discrete representation of the Earth and are preferable for the organization, integration, and analysis of large and multi-source geospatial datasets. Generating grids for the area of interest is usually the premise and basis for DGGS applications. Owing to incongruent hierarchies that restrict the multi-resolution applications of hexagonal DGGS, current grid generation of hexagonal DGGS for local areas mainly depends on inefficient single-resolution traversal methods by judging the spatial relationship between each cell and the area. This study designs a fast generation algorithm for local parts of hexagonal DGGS based on the hierarchical properties of DGGS. A partition structure at intervals of multiple levels is first designed to ensure the coverage relevance between parent and children cells of different levels. Based on this structure, the algorithm begins with coarser resolution grids and recursively decomposes them into the target resolution, with multiple decomposition patterns used and a unique condition proposed to make the generated grids without gaps or overlaps. Efficient integer coordinate operations are used to generate the vast majority of cells. Experimental results show that the proposed algorithm achieves a significant improvement in efficiency. In the aperture 4 hexagonal DGGS, the efficiency ratio of the proposed and traversal algorithms increases from six times in level 14 to approximately 339 times in level 18. This study provides a solid foundation for subsequent data quantization and multi-resolution applications in hexagonal DGGS and has broad prospects.

Discrete global grid systems (DGGS), which provide the reference framework for next-generation Digital Earth, recursively divide the Earth’s surface into discrete multi-resolution hierarchies and support multi-source spatial data fusion and location-related event simulation. Existing DGGS research has primarily focused on cell centers, and therefore lacks rigorous mathematical modeling of multi-structural elements that comprise centers, edges, and vertices, resulting in inefficient data exchange and inaccurate data modeling. To this end, a unified mathematical modeling method was proposed for hexagonal multi-structural elements of grid systems based on triangles. The code operation principles and indexing conversion relationships for multi-structural elements were also deduced. The proposed method provides a unified mathematical model which reveals the mathematical essence of hexagonal multi-structural elements and has theoretical significance for future research on hierarchical data structures. Furthermore, conversion between multi-structural elements of triangles and hexagons, was achieved with low algorithmic complexity. Finally, the use of multi-structural elements can improve the accuracy of data modeling and spatial analysis, which has important applications for the representation of geospatial location data.

Discrete Global Grid System (DGGS) is a new multi-resolution geospatial data modeling and processing scheme for the digital earth. The icosahedron is commonly regarded as an ideal polyhedron for constructing DGGSs with small distortions; however, the shape of its face is triangular, making it difficult to incorporate the matrix structure used for geospatial data storage and parallel computing. To overcome this limitation, this study utilizes the rhombic triacontahedron (RT) as the basic polyhedron to construct DGGSs. An equal-area projection between the surface of RT and the sphere is developed and used to design a grid-generation algorithm for the aperture 4 hexagonal DGGS based on RT. Compared with the equal-area DGGS based on the icosahedron, the proposed scheme results in smaller angular projection distortions, with the mean and standard deviation decreasing by 41.6% and 30.9%, respectively. The grid cells of the RT DGGS also achieve more optimized geometric characteristics in shape compactness, length deviation, and angle deviation than those in the icosahedron DGGS. Additionally, the cross-surface computation efficiency provides advantages in code conversion to latitude and longitude and proximity queries. Furthermore, the use of RT offers a new and better framework within the context of DGGS research and application.

六边形全球离散格网系统是支持多源地球空间信息融合处理的优选解决方案,相关研究已引起学术界广泛关注.相较于完整的全球格网,局部不规则区域格网的应用需求更加广泛,相关生成算法是当前六边形全球离散格网系统研究的重要问题之一.将二十面体相邻三角面组合为菱形逻辑结构,在此基础上,提出一种局部区域多孔径六边形格网系统生成算法.首先,分析格网剖分类型,建立离散整数坐标系,描述多孔径六边形格网单元空间位置;其次,将局部区域分解到球面二十面体的菱形逻辑结构上创建子区域;然后,根据子区域边界设计外接最小菱形遍历算法,剔除与 目标区域无关单元;最后,遍历最小菱形,生成多孔径局部格网.对比实验结果表明,提出的算法具有灵活性好、效率高等优点,生成的多孔径六边形格网用于栅格数据组织,可显著减少数据量,具有较好的应用潜力.

Hexagonal discrete global grids can provide an excellent solution for the massive multi-source, multi-temporal, and multi-resolution raster data integration and management. Traditional images are rectangular pixels, and cannot be expressed on a hexagonal grid. Therefore, how to obtain images based on hexagonal pixels has attracted widespread academic attention. Combining current hexagonal sampling methods, this paper studies the evaluation criteria of hexagonal sampling accuracy, summarizes the previous research, and proposes a more general hexagonal sampling method for remote sensing images. This method mainly involves signal preprocessing, spectrum analysis, calculation of sampling interval, and establishment of accuracy evaluation standards. Finally, we verify the feasibility of the proposed hexagon algorithm to provide a reference for hexagon sampling.

Discrete Global Grid Systems (DGGSs) are an emerging data model for integration and analysis of big Earth data. Current research on hexagonal DGGSs focuses mainly on the pure aperture, which limits application of resolution and may cause data redundancy. The mixed aperture hexagonal DGGS provides a new and flexible scheme that can optimize data precision by designing the most suitable grid resolution. This paper proposes a generic encoding and operation scheme for the mixed aperture 3 and 4 hexagonal DGGSs. A planar mathematical model with unique location representation was constructed. Based on this model, this paper designed an encoding scheme and gave relevant operational properties and rules. A generic method of extending the planar scheme to the surface of an icosahedron was then designed and spherical grids were obtained by projection. The conversion between geographical coordinates and codes was also provided. The proposed scheme has a more complete theoretical basis than existing schemes and can be applicable to different aperture sequences. Experiments show that the code addition efficiency of the proposed algorithm is superior to existing hexagonal grid systems. A case study with ocean salinity hexagonal data demonstrated the flexibility of the proposed scheme.

全球离散格网系统是数字化的多分辨率地球参考模型,在结构上支持多源位置相关信息的融合处理.基于正二十面体剖分的六边形全球离散格网系统具有较好的几何属性,相关研究已引起学术界的广泛关注,如何建立封闭球面上的六边形格网系统编码运算方案是当前的研究难点.研究表明,基于正多面体剖分的全球离散格网系统与正多面体格网系统拓扑等价,两者的编码运算结果也完全相同.根据这一原理,结合四孔六边形格网系统在正二十面体表面的分布特点,基于六边形格点四叉树定义顶点瓦片与面瓦片结构,提出了正二十面体四孔六边形格网系统编码运算方案.该方案通过高效编码运算实现了格网单元跨面操作,克服了现有成果需借助低效浮点数运算实现相同操作的缺陷.对比实验表明,该方案的跨面邻近单元搜索效率约是六边形四元平衡结构方案的19.6倍.

Discrete global grid systems (DGGSs) are an emerging multiresolution 3D model used to integrate and analyze big earth data. The characteristic of multiresolution is usually realized by hierarchically subdividing cells on the sphere using certain refinement. This paper introduces mixed aperture three- and four- icosahedral hexagonal DGGSs using two types of refinement, the various combinations of which can provide more resolutions compared with pure aperture hexagonal DGGSs and can flexibly design the aperture sequence according to the target resolutions. A general hierarchy-based indexing method is first designed, and related indexing arithmetics and algorithm are developed based on the indexing method. Then, the grid structure on the surface of the icosahedron is described and by projection spherical grids are obtained. Experiments show that the proposed scheme is superior to pure aperture schemes in choosing grid resolutions and can reduce the data volume by 38.5% in representing 1-km resolution raster dataset; using the proposed indexing arithmetics to replace spherical geometry operations in generating discrete spherical vector lines based on hexagonal cells can improve the generation efficiency.