Method for constructing equal-shape global discrete grid prism under spherical coordinate system
By constructing a spherical hexagonal mesh, the problems of grid consistency and stability in high-latitude areas under spherical coordinate systems are solved, and higher geophysical model accuracy and computational efficiency are achieved.
Patent Information
- Application Number
- CN202510046052.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-13
- Publication Date
- 2025-06-03
AI Technical Summary
Under spherical coordinate systems, it is difficult for the prior art to maintain grid consistency and stability in high latitude areas, resulting in discontinuity of geophysical models and reducing the accuracy of forward calculations.
By constructing a spherical regular hexagonal mesh, a closely adjacent global discrete mesh is generated using the aliquot points and rotation methods of spherical rhombus to ensure that the numbers of mesh along the symmetry axis direction of the spherical regular hexagonal mesh are equal.
The consistency and stability of the grid are achieved, the accuracy and computational efficiency of the geophysical model are improved, and the degradation problem of traditional grids in high latitude areas is overcome.
Smart Images
Figure CN120088426A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the fields of geophysics and spherical geometry technology. Specifically, it relates to a method for constructing prisms of equal-shaped global discrete grids in a spherical coordinate system. Background Art
[0002] In a spherical coordinate system, how to obtain accurate analytical solutions or numerical solutions for each unit has always been one of the core issues in forward modeling to select appropriate physical property units. In practical operations, similar to the point element method used for physical property inversion in a Cartesian coordinate system, when implementing physical property inversion in a spherical coordinate system, the common approach is to use radius, longitude, and latitude as basic parameters to subdivide the spherical shell into a series of independent discrete units. These units include but are not limited to point elements, line elements, surface elements, Prism units, Tesseroid units, combined forms of Prism and Tesseroid, and polyhedron units. Calculate the gravitational effect generated by each unit body, and then obtain the corresponding anomaly response through cumulative summation.
[0003] Regarding the spherical curvature fitting, taking the Prism unit as an example, although its gravitational field expression has an accurate analytical solution, when dealing with regional or global scale problems, it is necessary to fit the curvature by adjusting the distance and direction between Prism units. If the ground surface is too undulating, it will lead to discontinuities in the geophysical model, thereby reducing the accuracy of forward calculation and resulting in poor fitting effect of the spherical curvature. In contrast, the Tesseroid unit can accurately fit the spherical curvature, but it does not have a strict analytical solution for the gravity and magnetic potential fields and needs to rely on numerical integration methods to obtain the corresponding anomaly response.
[0004] However, when the spherical shell is discretized into Tesseroid units, the resulting series of units often lack consistency and may even exhibit structural degradation. For example, at the poles, the Tesseroid unit degenerates into a triangular prism; and the Tesseroid units divided at equal longitude and latitude intervals may also have spatial inhomogeneity. Taking geophysical gravity and magnetic forward modeling as an example, the core coefficient or kernel function of the physical property unit is affected by its volume effect. Due to the different shapes of the physical property units, it is difficult to reuse the core coefficient or kernel function, ultimately resulting in a significant increase in the forward calculation time. The above problems also exist for other physical property units such as triangular, rhombic prisms, or tetrahedral units.
[0005] Therefore, based on the proposed method for constructing a highly consistent spherical regular hexagon, the present invention uses adjacent equally divided points within three spherical rhombuses to construct new spherical equally divided points, and uses these equally divided points to construct closely adjacent global discrete grids (abbreviated as DGGS) and their units (i.e., prisms obtained from equal-shaped global discrete grids). Summary of the Invention
[0006] Problems to be Solved
[0007] This method constructs two spherical equilateral triangles (ΔABD, ΔCDB), combines them into a spherical rhombus ◇ABCD; uses the method of spherical rotation to obtain n θ equidistant points on each side (arc) of the spherical rhombus; selects equidistant points on two opposite sides (arcs) of the spherical rhombus (such as and ) to construct n θ arcs parallel to the other two sides of the spherical rhombus; and then uses the method of spherical rotation to obtain n θ equidistant points on the newly added arcs. At this time, based on the spherical points obtained above, a series of spherical regular hexagons can be obtained, with 6 spherical equidistant points as vertices and 1 spherical equidistant point as the center point. However, the number of these spherical regular hexagons along the symmetry axis direction of the spherical regular hexagon is not equal. Therefore, a new secondary equidistant point is constructed using three adjacent equidistant points. Subsequently, using six secondary spherical equidistant points as vertices and 1 spherical equidistant point as the center point, a spherical regular hexagon can be constructed. At this time, (n θ -2)×(n θ -2) closely adjacent spherical regular hexagons with equal numbers along the symmetry axis direction of the spherical regular hexagon can be obtained. Compared with the fact that the longitude and latitude grid changes with the longitude and latitude position, the spherical regular hexagons constructed by the present invention are of equal shape, and the spherical regular hexagon grid constructed by the present invention is "quasi"-square along the symmetry axis direction of the spherical regular hexagon, that is, the number is equal.
[0008] Technical Solution
[0009] To solve the above problems, the present invention adopts the following technical solutions.
[0010] A method for constructing an equi-shaped global discrete grid prism in a spherical coordinate system, comprising the following steps:
[0011] (1) Construct a spherical rhombus unit: Form a spherical rhombus unit by combining two spherical equilateral triangles to cover the research area;
[0012] (2) Generate equidistant points: Use the method of spherical rotation to equally divide each side of the spherical rhombus into a specified number of equidistant points to ensure that the central angle of the great circle between adjacent points is the same;
[0013] (3) Construct parallel arcs: Select equidistant points on two opposite sides of the spherical rhombus to generate multiple arcs parallel to the other two sides, and then use the method of spherical rotation to obtain newly added equidistant points;
[0014] (4) Generate the primary regular hexagon grid: Based on the above equally divided points, construct a regular hexagon grid with six spherical equally divided points as vertices and one spherical equally divided point as the center point.
[0015] (5) Optimize grid consistency: To solve the problem of uneven quantity of the primary regular hexagon grid along the symmetry axis direction, generate sub - level equally divided points using three adjacent equally divided points, and construct new regular hexagon units accordingly.
[0016] (6) Form the final grid: Through recursive subdivision and optimization, generate a globally discrete regular hexagon grid with uniform quantity and consistent shape along the symmetry axis direction.
[0017] Preferably, in step (1), use part of the spherical surface as the discrete object, that is, by constructing two spherical equilateral triangles ΔABD / ΔCDB and combining them into a spherical rhombus ABCD to cover the study area.
[0018] Preferably, for the spherical rhombus ◇ABCD in step (2), select the specific form of equally divided points, construction direction, and number of equal divisions n θ .
[0019] Preferably, for the construction form, construction direction, and number of equal divisions n in step (3) θ , use the method of spherical rotation to discretize each side of the spherical rhombus ◇ABCD into n θ - 1 equal divisions, that is, obtain n θ equally divided points; to obtain multiple equally divided points of the arc from u 0 (the vector from the center of the sphere to the starting point of the arc to be equally divided) to u 1 (the vector from the center of the sphere to the ending point of the arc to be equally divided) on the circular arc, use the method of spherical linear interpolation, and the equally spaced point u t can be written as u 0 and u t are the starting point and ending point that form the corresponding circular arc, u 0 is the vector from the center of the sphere to the starting point u 0 of the arc to be equally divided, the vector from the center of the sphere to the ending point u t of the arc to be equally divided, θ is the central angle of the arc to be equally divided corresponding to the center of the sphere, and t corresponds to the ratio of the central angle formed by u 0 and u t to the central angle formed by u 0 and u 1 .
[0020] Preferably, in step (4), according to the construction form, construction direction, and number of equally divided points n θ , select a pair of opposite sides of the spherical rhombus ◇ABCD, and construct n θ mutually parallel arcs corresponding to a group of great circular arcs one by one.
[0021] Preferably, in step (5), according to n θ mutually parallel arcs, and then by using the method of spherical rotation, n θ equidistant points of each newly added arc are obtained, and a total of equidistant points are obtained.
[0022] Preferably, in step (6), according to the obtained equidistant points, a series of spherical regular hexagons are obtained, which are formed by six spherical equidistant points as vertices and one spherical equidistant point as the center point.
[0023] Preferably, in step (7), according to the fact that the numbers of the obtained spherical regular hexagons along the symmetry axis directions of the spherical regular hexagons are not equal, by using three adjacent equidistant points, a new sub-level equidistant point is constructed. Assume that there is a point H between any three adjacent equidistant points E, F, and G, and accordingly, a coplanar equation is constructed as follows:
[0024]
[0025] In the formula, (x α , y α , z α ), α ∈ [E, F, G, H] are the coordinate values of the three adjacent equidistant points E, F, and G in the geocentric coordinate system,
[0026] Preferably, in step (8), a spherical regular hexagon is constructed with six sub-level spherical equidistant points as vertices and one spherical equidistant point as the midpoint.
[0027] Advantageous Effects
[0028] Compared with the prior art, the advantageous effects of the present invention are as follows:
[0029] Overcoming the limitations of the longitude and latitude grid: The traditional longitude and latitude grid will degenerate in high-latitude regions (especially near the poles), resulting in inconsistent grid shapes and affecting the accuracy of data. The present invention solves this problem by constructing a spherical regular hexagon grid with equal shapes, ensuring the consistency and stability of the grid.
[0030] Improving the calculation accuracy and efficiency: By generating multi-level grids with consistent shapes and continuous scales, the present invention realizes seamless coverage globally. This consistency enables the kernel coefficients or kernel functions of physical property units to be reusable, thus significantly reducing the time and storage requirements for geophysical forward modeling calculations.
[0031] Enhanced applicability and flexibility: The method of the present invention only depends on the four endpoints of the spherical rhombus and fractional arc lengths, etc. The algorithm is simple and efficient, and is applicable to the coverage requirements of different research areas. At the same time, by using three adjacent equally divided points to construct a new sub-level equally divided point, the grid structure can be further optimized.
[0032] Improved spherical curvature fitting effect: Compared with traditional prismatic elements or Tesseroid elements, the equiform spherical regular hexagon grid generated by the present invention can better fit the spherical curvature, avoiding the discontinuity problem caused by surface undulations and improving the model accuracy.
[0033] Adapted to multi-resolution spatial data processing: The global discrete grid (DGGS) framework generated by the present invention has excellent multi-resolution characteristics, can effectively express spatial position, scale and accuracy, and is an ideal tool for processing geophysical problems at the regional or global scale.
[0034] In summary, the present invention provides strong technical support for geophysical research and spatial data processing by constructing a consistent, stable and efficient spherical regular hexagon grid. Brief Description of the Drawings
[0035] Figure 1 is a flowchart of the method for constructing an equiform global discrete grid prism in a spherical coordinate system according to the present invention;
[0036] Figure 2 is a schematic diagram of the spherical rhombus of the present invention; (a) construction schematic diagram, (b) schematic diagram of covering the research area;
[0037] Figure 3 is a schematic diagram of the survey network construction form and direction of the present invention; (a) upright regular hexagon, (b) horizontal regular hexagon;
[0038] Figure 4 is a schematic diagram of quaternion interpolation according to the present invention;
[0039] Figure 5 is a schematic diagram of constructing parallel arcs at the equally divided points of the spherical rhombus of the present invention;
[0040] Figure 6 is a schematic diagram of the equally divided points of the spherical rhombus of the present invention;
[0041] Figure 7 is an example diagram of constructing a spherical regular hexagon based on the equally divided points of the spherical rhombus of the present invention; (a) horizontal regular hexagon, (b) vertical regular hexagon;
[0042] Figure 8 is a schematic diagram of constructing a new equally divided point by using three adjacent equally divided points according to the present invention;
[0043] Figure 9This is an example diagram of a regular hexagonal sphere centered at three adjacent equally divided points of the present invention. Detailed implementation mode
[0044] The present invention will be further described below in conjunction with specific embodiments.
[0045] Implementation plan
[0046] As Figure 1 shown, a method for constructing a prism of a global discrete grid of equal shape in a spherical coordinate system includes the following steps:
[0047] As Figure 2 shown, compared with the tesseroid unit in high-latitude (>85°) regions, especially when approaching the poles, where degradation occurs, the global discrete grid (DGGS) recursively generates multi-level grids with similar shapes and continuous scales by subdividing regular polyhedron patches and projecting them onto the sphere, achieving seamless and stable full coverage, overcoming the limitations of the longitude and latitude grid, effectively expressing spatial position, scale, and accuracy, and is an excellent framework for global multi-resolution spatial data processing. However, affected by the projection method, it is easy to cause the units on the sphere (after projection) to be difficult to maintain the equal-area or equal-shape characteristics of the units on the original plane (before projection). Therefore, only a part of the sphere is used as the discrete object, that is, by constructing two spherical equilateral triangles (ΔABD, ΔCDB) and combining them into a spherical rhombus ◇ABCD to cover the study area.
[0048] For the spherical rhombus ◇ABCD, as Figure 3 shown, select a specific equally divided point construction form (i.e., upright regular hexagon or horizontal regular hexagon), construction direction (i.e., a direction vector starting from the center point of the regular hexagon and passing through the midpoint of a certain side of the regular hexagon), and the number of equal divisions n θ .
[0049] The construction form, construction direction, and the number of equal divisions n θ , using the method of spherical rotation (as Figure 4 shown) to discretize each side of the spherical rhombus ◇ABCD into n θ -1 equal divisions, that is, obtaining n θ equally divided points. That is, since the central angles of the great circles corresponding to the arcs formed by two adjacent points are the same, therefore, one of the equally divided parallel lines (arcs) to be constructed can be placed on the following plane, that is, the tangent plane of the great circle of the unit sphere.
[0050] To obtain u 0 (vector from the center of the sphere to the starting point of the arc to be equally divided) to u 1For multiple equally-spaced points on an arc of the vector from the center of the sphere to the end point of the arc to be equally divided, using the Spherical Linear Interpolation method, equally-spaced points u t can be written as where θ is the central angle of the sphere corresponding to the arc to be equally divided, and t corresponds to u 0 and u t the ratio of the central angle formed by and u 0 and u 1 to the central angle formed by and u
[0051] For the described construction form, construction direction, and number of equally-spaced points n θ , select a set of opposite sides (here two great arcs) of the described spherical rhombus ◇ABCD. For a set of great arcs (involving 2n θ equally-spaced points), construct n θ mutually parallel arcs one by one.
[0052] For the described n θ mutually parallel arcs ( Figure 5 ), then using the method of spherical rotation ( Figure 4 ), obtain the n θ equally-spaced points of each newly added arc, that is, a total of equally-spaced points ( Figure 6 ).
[0053] For the obtained equally-spaced points, a series of spherical regular hexagons can be obtained with 6 spherical equally-spaced points as vertices and 1 spherical equally-spaced point as the center point.
[0054] The number of these obtained spherical regular hexagons along the axis of symmetry of the spherical regular hexagon is not equal ( Figure 7 ). For this reason, using three adjacent equally-spaced points, construct a new sub-level equally-spaced point ( Figure 8 ). That is, assume there is a point H between any three adjacent equally-spaced points E, F, and G. Based on this, a coplanar equation can be constructed as follows:
[0055]
[0056] In the formula, (x α , y α , z α ), α ∈ [E, F, G, H] are the coordinate values of the three adjacent equally-spaced points E, F, and G in the geocentric coordinate system,
[0057] Subsequently, using 6 sub-level spherical equally-spaced points as vertices and 1 spherical equally-spaced point as the midpoint, construct a spherical regular hexagon. When n θ = 7, for exampleFigure 9 As shown, the number of (n θ -2)×(n θ -2) spherical regular hexagons with equal numbers in the direction of the axis of symmetry of the spherical regular hexagon can be obtained. In the figure, the center of the spherical regular hexagon is marked with — number / number — corresponding to the node number of the spherical equal division point / spherical regular hexagon number respectively.
[0058] Comprehensively Figure 2-9 , the present invention shows broad prospects from the application perspective, which are mainly reflected in the following aspects:
[0059] (1) High-precision modeling in the geophysical field
[0060] By constructing global discrete grid prisms of equal shapes, the present invention overcomes the problem of degradation of traditional latitude-longitude grids and Tesseroid cells in high-latitude regions. This consistency and stability make it superior in geophysical forward and inverse calculations, especially when dealing with regional or global-scale problems, it can improve the curvature fitting accuracy and reduce calculation errors.
[0061] (2) Multi-resolution spatial data processing
[0062] The global discrete grid (DGGS) framework has a multi-level structure and seamless coverage characteristics, and can be used for the expression and analysis of multi-resolution spatial data. This ability makes it of important application value in geographic information systems (GIS), and can support spatial data processing, storage and analysis on a global scale.
[0063] (3) Efficient calculation and storage optimization
[0064] Since the grid cells generated by the present invention have consistent shapes and can be reused, its kernel coefficients or kernel functions are regular in geophysical models, greatly reducing the storage requirements and calculation time. This is particularly important for large-scale data processing (such as the ETOPO1 model), and can significantly improve the calculation efficiency.
[0065] (4) Modeling suitable for complex terrains
[0066] The present invention better fits the spherical curvature through regular hexagon grids, avoiding the discontinuity problem caused by surface undulations, and is suitable for the modeling and analysis of complex terrains.
[0067] (5) Wide interdisciplinary applications
[0068] In addition to geophysics, this method can also be applied to fields such as meteorology, oceanography, and ecology that require global discrete grid support. For example, it can be used for climate simulation, ocean hydrodynamic analysis, or ecosystem modeling, etc.
[0069] (6) Flexible adaptation to different research areas
[0070] By selecting specific grid construction forms, directions, and equal division numbers, the present invention can flexibly adapt to the requirements of different research areas. This flexibility enables it to play a role in both local and global research.
[0071] In summary, the present invention not only solves the degradation problem of traditional grids in high-latitude regions but also provides an efficient, accurate, and flexible global discrete grid construction method for multiple fields, with broad application prospects.
[0072] The above content further elaborates on the present invention in combination with specific implementation manners. It cannot be determined that the specific implementation of the present invention is only limited to these descriptions. For those of ordinary skill in the geophysical field and spherical geometry technology field to which the present invention pertains, without departing from the concept of the present invention, several simple deductions or substitutions can still be made, which should all be regarded as belonging to the protection scope determined by the claims submitted for the present invention.
Claims
1. A method for constructing a global discrete grid prism of equal shape in a spherical coordinate system, characterized by: The following steps are involved: (1) Constructing a spherical rhombus unit: By merging two spherical regular triangles, a spherical rhombus unit is formed to cover the study area; (2) Generate equally divided points: Use the spherical rotation method to divide each side of the spherical rhombus into a specified number of equally divided points, ensuring that the central angles of the great circles between adjacent points are consistent; (3) Constructing parallel arcs: Selecting equal points on two opposite sides of a spherical rhombus, generating multiple arcs parallel to the other two sides, and then using the spherical rotation method to obtain newly added equal points; (4) Generating a primary regular hexagonal grid: Based on the above-mentioned equally divided points, a regular hexagonal grid is constructed with six equally divided points on the sphere as vertices and one equally divided point on the sphere as the center point; (5) Optimizing grid consistency: To solve the problem of uneven number of primary regular hexagonal grids along the symmetry axis, three adjacent equal-division points are used to generate secondary equal-division points, and new regular hexagonal units are constructed based on them. (6) Forming the final grid: Through recursive subdivision and optimization, a global discrete regular hexagonal grid with uniform number and consistent shape along the symmetry axis is generated.
2. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 1, characterized in that: In step (1), part of the sphere is used as a discrete object, that is, by constructing two spherical equilateral triangles ΔABD / ΔCDB and merging them into a spherical rhombus ABCD to cover the study area.
3. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 2, characterized in that: The spherical rhombus ◇ABCD described in step (2) is constructed by selecting the specific equal-division point construction form, construction direction and equal-division number n. θ .
4. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 3, characterized in that: The construction form, construction direction and equal number n in step (3) θ , discretize the sides of the spherical rhombus ◇ABCD into n using the spherical rotation method θ -1 is divided equally, that is, n θ To obtain multiple equally divided points from u0 to u1 on the arc, the spherical linear interpolation method is used, and the equally spaced points u t Written as u0 and u t The starting point and end point of the corresponding arc are formed. u0 is the vector from the center of the sphere to the starting point u0 of the arc to be equally divided, and u t vector, θ is the corresponding spherical center angle of the arc to be divided equally, and t corresponds to u0 and u t The ratio of the spherical center angle formed by u0 and u1 to the spherical center angle formed by u0 and u1.
5. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 4, characterized in that: In step (4), the structural form, structural direction and number of equally divided points n are θ , select a pair of opposite sides of the spherical rhombus ◇ABCD, and construct n corresponding to a pair of great circle arcs. θ Parallel arcs.
6. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 5, characterized in that: In step (5), according to n θ arcs parallel to each other, and then use the method of spherical rotation to obtain the n of each newly added arc θ Equal points, a total of Equal points.
7. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 6, characterized in that: In step (6), according to the obtained points of equal division, we obtain a series of spherical regular hexagons consisting of 6 spherical equal division points as vertices and one spherical equal division point as the center point.
8. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 7, characterized in that: In step (7), according to the fact that the number of the obtained spherical regular hexagons along the symmetry axis of the spherical regular hexagon is not equal, a new second-level equal division point is constructed using three adjacent equal division points. Assuming that there is a point H between any three adjacent equal division points E, F and G, a coplanar equation is constructed as follows: In the formula, (x α ,y α ,z α ), α∈[E,F,G,H] is the coordinate value of three adjacent equally divided points E, F and G in the geocentric coordinate system, 9. The method for constructing a global discrete grid prism of equal shape in a spherical coordinate system according to claim 8, characterized in that: In step (8), a spherical regular hexagon is constructed using six secondary spherical dividing points as vertices and one spherical dividing point as a midpoint.