Search Results (18)

In this article, we introduce analogues of classic Euclidean bounds, including spherical caps, geodesic axis-aligned bounding boxes (AABBs), geodesic oriented bounding boxes (OBBs), and geodesic k-discrete oriented polytopes (k-DOPs). We also formulate k-circle coverage, a union of variable-radius caps solved by a binary integer program over candidates generated from Discrete Global Grid System (DGGS)-based rasterization. As all constructions run directly on the spherical surface, $S^2$, they preserve geodesic distances and avoid projection distortion. We benchmark these methods on seven country boundary polygons consisting of thousands of points, and report construction time, memory, tightness, and query throughput. Results show our analytic geodesic bounds deliver orders of magnitude improvements over exact tests, with trade-offs in tightness: spherical caps are fastest but loosest; geodesic OBBs are a strong balance; geodesic k-DOPs consistently have the tightest bounds. k-circle coverage has spherical cap query speed while also having locally adaptive fits; construction time increases with DGGS resolution. Altogether, these bounds specific to the sphere provide practical, conservative filters for globe-scale Digital Earth queries. Full article

Agile Earth Observation Satellites (AEOSs), with their maneuverability, can flexibly observe point, line and region targets. However, existing research typically requires distinct algorithms for each target type, lacking a unified modeling and solution framework, which hinders the ability to meet the demands of rapid and coordinated observation of multiple target types in complex scenarios. To address these issues, this paper proposes a unified scheduling model for agile Earth observation satellites based on the Degenerate Quadtree Grid (DQG) and Proximal Policy Optimization (PPO), termed AEOSSP-USM. Firstly, the DQG is first employed to enable unified management and integrated modeling of point, line, and area targets; Secondly, traditional time window calculations based on longitude and latitude are replaced with grid code-based computations using DQG; Finally, the PPO algorithm, a deep reinforcement learning method, is introduced to formulate AEOSSP-USM as a Markov Decision Process (MDP), enabling efficient problem solving. Experimental results demonstrate that the proposed method effectively realizes unified scheduling of heterogeneous targets, improving imaging quality about 3 times, reducing energy consumption by 10%, decreasing memory usage more than 90%, and enhancing computational efficiency by 35 times compared to conventional longitude-latitude strip algorithm. Full article

This research introduces an innovative probabilistic method for examining torsional stress behavior in spherical shell structures through Monte Carlo simulation techniques. The spherical geometry of these components creates distinctive computational difficulties for conventional analytical and deterministic numerical approaches when solving torsion-related problems. The authors develop a comprehensive mesh-free Monte Carlo framework built upon the Feynman–Kac formula, which maintains the geometric symmetry of the domain while offering a probabilistic solution representation via stochastic processes on spherical surfaces. The technique models Brownian motion paths on spherical surfaces using the Euler–Maruyama numerical scheme, converting the Saint-Venant torsion equation into a problem of stochastic integration. The computational implementation utilizes the Fibonacci sphere technique for achieving uniform point placement, employs adaptive time-stepping strategies to address pole singularities, and incorporates efficient algorithms for boundary identification. This symmetry-maintaining approach circumvents the mesh generation complications inherent in finite element and finite difference techniques, which typically compromise the problem’s natural symmetry, while delivering comparable precision. Performance evaluations reveal nearly linear parallel computational scaling across up to eight processing cores with efficiency rates above 70%, making the method well-suited for multi-core computational platforms. The approach demonstrates particular effectiveness in analyzing torsional stress patterns in thin-walled spherical components under both symmetric and asymmetric boundary scenarios, where traditional grid-based methods encounter discretization and convergence difficulties. The findings offer valuable practical recommendations for material specification and structural design enhancement, especially relevant for pressure vessel and dome structure applications experiencing torsional loads. However, the probabilistic characteristics of the method create statistical uncertainty that requires cautious result interpretation, and computational expenses may surpass those of deterministic approaches for less complex geometries. Engineering analysis of the outcomes provides actionable recommendations for optimizing material utilization and maintaining structural reliability under torsional loading conditions. Full article

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. Full article

This paper presents a novel Earth-System Stratified Grid (ISEA4H-ESSG) model, designed to address the challenges in multi-layer geoscience data management and analysis. In the realm of geosciences, which encompasses the solid earth, atmosphere, hydrosphere, and biosphere, as well as planetary and space sciences, the effective integration of diverse data sources is crucial. Traditional grids have limitations in three-dimensional spatial modeling, cross-layer data fusion, and dynamic multi-scale analysis. The ISEA4H-ESSG model overcomes these drawbacks by integrating the Icosahedral Snyder Equal-Area Aperture 4 Hexagon Discrete Global Grid System (ISEA4H DGGS) with a degenerative subdivision mechanism. It adheres to six core principles, including stratified spherical coverage, geographic consistency, multi-scale dynamic adaptability, global seamless partitioning, encoding uniqueness and efficiency, and multi-source data compatibility. Through the independent subdivision of spherical and radial layers, this model balances resolution differences and resolves polar-grid distortion and cross-layer data heterogeneity issues. The introduction of a four-dimensional spatiotemporal encoding framework enhances the storage and parallel computing capabilities of massive datasets. Case studies on ionosphere three-dimensional modeling and global atmospheric temperature field formatting demonstrate the high precision and adaptability of the ISEA4H-ESSG model. This research provides a unified spatial data infrastructure for geosciences, facilitating in-depth studies on natural hazards, climate change, and planetary evolution, and offering new perspectives for international partnerships and future Earth-related research. Full article

The Discrete Global Grid System (DGGS) provides a foundational framework for the digital Earth, where uniform intercell distances are essential for accurate numerical simulations. However, due to the spherical topology, achieving strictly equidistant spherical grid cells is impractical. Most existing studies have focused on regional scales, which are constrained by data acquisition limitations and render global equidistant optimization algorithms economically infeasible. The equidistant characteristics of cells are influenced by map projections and often exhibit regional variations. In this paper, we analyze these equidistant characteristics and construct an equidistant pattern for an icosahedral hexagonal DGGS. By integrating this pattern into the icosahedral orientation method, we develop a regional-scale equidistant optimization method for DGGS. Experiments on river network extraction in the Yangtze River Basin demonstrate significant improvements: the equidistance of grid cells covering the region increased by over 34.2%, while the accuracy of river network extraction improved by 51.41%. Moreover, this method is extensible to other grid models, enhancing the broader applicability of DGGS. Full article

Recent advancements in geospatial technologies have significantly expanded the volume and diversity of geospatial data, unlocking new and innovative applications that require novel Geographic Information Systems (GIS). (Discrete) Global Grid Systems (DGGSs) have emerged as a promising solution to further enhance modern geospatial capabilities. Current DGGSs employ a simple, low-resolution polyhedral approximation of the Earth for efficient operations, but require a projection between the Earth’s surface and the polyhedral faces. Equal-area DGGSs are desirable for their low distortion, but they fall short of this promise due to the inefficiency of equal-area projections. On the other hand, efficiency-first DGGSs need to better address distortion. We introduce a novel mesh-based DGGS (MBD) which generalizes efficient operations over watertight triangular meshes with spherical topology. Unlike traditional approaches that rely on Platonic or Catalan solids, our mesh-based method leverages high-resolution spherical meshes to offer greater flexibility and accuracy. MBD allows high-resolution polyhedra (HRP) to be used as the base polyhedron of a DGGS, significantly reducing distortion. To address the operational challenges, we introduce a new hash encoding method and an efficient barycentric indexing method (BIM). MBD extends Atlas of Connectivity Maps to the BIM to provide efficient spatial and hierarchical traversal. We introduce several new base polyhedra with lower areal and angular distortion, and we experimentally validate their properties and demonstrate their efficiency. Our experimentation shows that we achieve constant-time operations for high-resolution MBD, and we recommend polyhedra to be used as the base polyhedron for low-distortion DGGSs, compact faces, and efficient operations. Full article

Earth system simulation technology is fundamental for ecological protection and high-quality development in the Yellow River Basin. To address the lack of a Yellow River simulation platform, this study proposes an adaptive multiscale true 3D crust simulation platform using the Sphere Geodesic Octree Grid (SGOG). Twelve models in four categories were designed: single fine-scale models, geomorphic zone-based models, and models using both top-down and bottom-up approaches. The models were evaluated based on terrain feature representation and computational efficiency. The results show that single fine-scale models preserve detailed terrain features but are computationally intensive. They are suitable for the precise simulation of surface processes. Top-down and bottom-up models balance terrain detail and efficiency, and are thereby widely applicable. Geomorphic zone-based models provide detailed focal area representation and higher computational efficiency, being more targeted. Various methods offer flexible scale transformations, each with its own strengths, allowing researchers to select a method according to practical application needs. Consequently, this research demonstrates that spherical discrete grids offer reliable support for constructing basin simulation platforms, providing new technological and scientific insights for the Yellow River Basin’s ecological protection and development. Full article

For regional or even global geophysical problems, the curvature of the geophysical model cannot be approximated as a plane, and its curvature must be considered. Tesseroids can fit the curvature, but their shapes vary from almost rectangular at the equator to almost triangular at the poles, i.e., degradation phenomena. Unlike other spherical discrete grids (e.g., square, triangular, and rhombic grids) that can fit the curvature, the Discrete Global Grid System (DGGS) grid can not only fit the curvature but also effectively avoid degradation phenomena at the poles. In addition, since it has only edge-adjacent grids, DGGS grids have consistent adjacency and excellent angular resolution. Hence, DGGS grids are the best choice for discretizing the sphere into cells with an approximate shape and continuous scale. Compared with the tesseroid, which has no analytical solution but has a well-defined integral limit, the DGGS cell (prisms obtained from DGGS grids) has neither an analytical solution nor a fixed integral limit. Therefore, based on the isoparametric transformation, the non-regular DGGS cell in the system coordinate system is transformed into the regular hexagonal prism in the local coordinate system, and the DGGS-based forwarding algorithm of the gravitational field is realized in the spherical coordinate system. Different coordinate systems have differences in the integral kernels of gravity fields. In the current literature, the forward modeling research of polyhedrons (the DGGS cell, which is a polyhedral cell) is mostly concentrated in the Cartesian coordinate system. Therefore, the reliability of the DGGS-based forwarding algorithm is verified using the tetrahedron-based forwarding algorithm and the tesseroid-based forwarding algorithm with tiny tesseroids. From the numerical results, it can be concluded that if the distance from observations to sources is too small, the corresponding gravity field forwarding results may also have ambiguous values. Therefore, the minimum distance is not recommended for practical applications. Full article

In this paper, the derivation of a concise closed form for the gravitational field of a polyhedron is presented. This formula forms the basis of the algorithm for calculating the gravitational field of an arbitrary shape body with high accuracy. Based on this algorithm, a method for gravity data inversion (creating density models of the Earth’s crust) has been developed. The algorithm can accept either regular or irregular polyhedron discretization for density model creation. The models are approximated with dense irregular grids, elements of which are polyhedrons. When performing gravity data inversion, we face three problems: topography with large amplitude, the sphericity of the planet, and a long computation time because of the large amount of data. In our previous works, we have already considered those problems separately but without explaining the details of the computation of the closed-form solution for a polyhedron. In this paper, we present for the first time a performance-effective numerical method for the inversion of gravity data based on topography. The method is based on closed-form expression for the gravity field of a spherical density model of the Earth’s crust with the upper topography layer, and provides great accuracy and speed of calculation. There are no restrictions on the model’s geometry or gravity data grid. As a case study, a spherical density model of the Earth’s crust of the Urals is created. Full article

Numerical seismic wave field simulation is essential for studying the dynamic responses in semi-infinite space, and the absorbing boundary setting is critical for simulation accuracy. This study addresses spherical waves incident from the free boundary by applying dynamic equations and Rayleigh damping. A new multi-directional viscous damping absorbing boundary (MVDB) method is proposed based on regional attenuation. An approximate formula for the damping value is established, which can achieve absorbing the boundary setting by only solving the mass damping coefficients without increasing the absorbing region grid cells or depending on the spatial and temporal walking distance. The validity and stability of the proposed method are proven through numerical calculations with seismic sources incident from different angles. Meanwhile, the key parameters affecting the absorption of the MVDB are analyzed, and the best implementation scheme is provided. In order to meet the requirements of mediums with different elastic parameters for boundary absorption and ensure the high efficiency of numerical calculations, the damping amplitude control coefficients k can be set between 1.02 and 1.12, the thickness of the absorbing region L is set to 2–3 times of the wavelength of the incident transverse wave, and the thickness of the single absorbing layer is set to the size of the discrete mesh of the model Δl. Full article

We present a new explicit quasi-Lagrangian integration scheme with the three-dimensional cubic spline function transform (transform = fitting + interpolation, referred to as the “spline format”) on a spherical quasi-uniform longitude–latitude grid. It is a consistent longitude–latitude grid, and to verify the feasibility, accuracy, convergence, and stability of the spline format interpolation scheme for the upstream point on the longitude–latitude grid, which may map a quasi-uniform longitude–latitude grid, a set of ideal, exact test schemes is adopted, which are recognized and proven to be effective internationally. The equilibrium flow test, cross-polar flow test, and Rossby–Haurwitz wave test are used to illustrate the spline scheme uniformity to the linear scheme and to overcome the over-dense grid in the polar region and the non-singularity of the poles. The cross-polar flow test demonstrates that the geostrophic wind crosses the polar area correctly, including the South Pole and North Pole. A non-hydrostatic, fully compressible dynamic core is used to complete the density flow test, demonstrating the existence of a time-varying reference atmosphere and that the spline format can simulate highly nonlinear fine-scale transient flows. It can be compared for the two results of the density flow test between the solution with the spline format and the benchmark reference solution with the linear format. Based on the findings, the non-hydrostatic dynamic core with the spline format is recommended for adoption. When simulated for the flow over an ideal mountain, through the “topographic gravity wave test”, the bicubic surface terrain and terrain-following height coordinates, time-split integration, and vector discrete decomposition can be derived successfully. These may serve as the foundations for a global, unified spline-format numerical model in the future. Full article

With the continuous development of general aviation, the contradiction between the air demand of general aviation low-altitude airspace and civil aviation routes is sharp. The difficulty of airspace planning is complex and changeable, and the existing working mode of simply using computer mapping and manually finding airspace conflict contradictions can no longer meet the large-scale air use demand. In response to the existing spatial representation model of longitude and latitude grid, which has large grid deformation in high latitude areas, and the problem of slow computation speed of the conflict detection (CD) algorithm that determines whether the airspace boundary coordinates overlap, we propose a grid model that represents airspace with a spherical rhombic discrete grid of positive icosahedron and design a matrix-based digital representation method of airspace, which uses matrix product operation. The matrix product operation is used to quickly determine whether there is a conflict between airspace and airspace and between airspace and routes. Full article

DGGS (Discrete Global Grid System) has many subdivision models and coding methods. Due to the lack of underlying consistency of different DGGS codes, most of them are converted through longitude–latitude, which greatly reduces the interoperability efficiency of different DGGS data and has become one of the bottlenecks in efficient integration of multi-source DGGS data. Therefore, a direct mapping method from one grid code to another (Grid to Grid, GtoG) for multi-type DGGSs is proposed based on three classical DGGSs (triangular, diamond and hexagonal grids) and two commonly used filling curves (Hilbert curve and Z-curve). The mutual conversion rules of different grids expressing spatial point, line and surface data are constructed. Then, the above method is extended to the spherical icosahedral grid framework, and three different region coding mapping rule tables of the basic inside cells, boundary cells and vertex cells are designed. Finally, the experimental results show that, compared with the longitude–latitude conversion method, the average conversion efficiency of spatial point, line and surface data is increased by 2–4 orders of magnitude. This new method greatly improves the interoperability efficiency and provides a feasible solution for the efficient integration of multi-source DGGS data. Full article

Ocean modeling and simulation are important for understanding the dynamic processes in the geophysical system, and the simulation of tidal dynamics is of great significance for understanding the dynamic evolution of the ocean. However, there are some problems in existing simulations, including lack of specific standards to produce a desirable discrete spherical mesh for global ocean modelling. Many global ocean numerical models based on conventional longitude-latitude (LL) coordinates suffer from the “pole problem” in regions adjacent to the North Pole due to the convergence of meridians, which seriously hinders global ocean simulations. In this paper, a new longitude-latitude spherical grid coupled with rotated coordinate mapping is proposed to overcome the problem. In the design of the numerical model, for spatial approximation, the finite volume method on staggered C grid is proposed to solve the two-dimensional tidal wave equations for the global ocean. For temporal integration, the third-order Adams-Bashforth method is used to explicitly extrapolate the value on the next time interval half layer, and then the fourth-order implicit Adams-Moulton method is used to correct the water level. Finally, the constructed model is used to simulate the dynamics of two-dimensional tidal waves in the global ocean, and the co-tidal maps of two major diurnal tide and semidiurnal tide components are shown. The results demonstrate that the proposed model can support the simulation of tidal dynamics in the global ocean, especially for the Arctic Ocean. Full article