CN115857036B - Application of novel spherical surface uniform distribution grid in spherical harmonic analysis - Google Patents
Application of novel spherical surface uniform distribution grid in spherical harmonic analysis Download PDFInfo
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Abstract
The invention discloses an application of a novel spherical uniform distribution grid in spherical harmonic analysis, which comprises the following steps: step 1: according to a selection method of HEALPix basic grids, generating 20 basic grids which are symmetrical in terms of north-south hemispheres on a sphere by combining with a Plastria regular icosahedron structure, and continuously subdividing each basic grid to obtain a novel spherical uniform distribution grid ico _HEALPix; step 2: combining the isosurface integral cloth characteristics of the ico _HEALPix grid, and determining the spherical coordinates at each grid point under different resolutions; step 3: and (3) adopting an iterative block diagonal least square method to realize the spherical harmonic analysis under the ico _HEALPix grid. Compared with HEALPix, the ico-HEALPix grid provided by the invention has higher data processing precision in a high-latitude area, and solves the problem of high-latitude data redundancy under the condition of no precision loss compared with the traditional geographic grid.
Description
Technical Field
The invention belongs to the technical field of physical geodetic measurement, and particularly relates to application of a novel spherical surface uniform distribution grid in spherical harmonic analysis.
Background
Traditional ground gravity measurement planning, data statistics, data processing, gravity field model construction and the like are all carried out in a geographic grid distribution mode, and along with continuous improvement of gravity data resolution and corresponding data volume, the traditional geographic grid distribution brings a plurality of limitations, and particularly, data redundancy exists in high-latitude areas. In order to solve the defects of the mass data processing and sampling in the distribution mode, gorski teaches a multifunctional structure HEALPix to NASA, and is successfully applied to data processing and analysis of universe microwave background (CMB, cosmic Microwave Background) experiments to generate a high-resolution CMB space diagram, thereby playing an important role in improving the precision and efficiency of data management, data processing, spherical harmonic analysis and the like.
Disclosure of Invention
Aiming at the problem that in the ball resonance analysis process by utilizing the HEALPix grid, the HEALPix grid has larger error in the ball resonance analysis result due to insufficient data sampling in a high latitude area, the invention provides the ico-HEALPix (icosahedral HEALPix) grid, and the data sampled by the grid not only effectively improves the utilization rate of the data in the ball resonance analysis process and the stability of the ball resonance analysis structure, but also solves the problem that the ball resonance analysis precision of the HEALPix grid is poor due to insufficient data sampling in the high latitude area.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
an application of a novel spherical uniform distribution grid in spherical harmonic analysis, comprising:
step 1: according to a selection method of HEALPix basic grids, generating 20 basic grids which are symmetrical in terms of north-south hemispheres on a sphere by combining with a Plastria regular icosahedron structure, and continuously subdividing each basic grid to obtain a novel spherical uniform distribution grid ico _HEALPix;
step 2: combining the isosurface integral cloth characteristics of the ico _HEALPix grid, and determining the spherical coordinates at each grid point under different resolutions;
step 3: and (3) adopting an iterative block diagonal least square method to realize the spherical harmonic analysis under the ico _HEALPix grid.
Further, the step 1 includes:
expanding total 12 basic grids of HEALPix 3×4 to total 20 basic grids of 4×5, rotating the entire southern hemisphere of the basic grid of 4×5 by 36 ° to obtain grids symmetrical about the equator north and south, and subdividing each basic grid into N in each direction side ＝2 k Segments and form a sub-grid, where N side And k is a parameter representing resolution.
Further, the step 1 further includes:
the characteristics of data hierarchical construction, area and latitude distribution of the HEALPix grid are reserved.
Further, the step 2 includes:
the whole sphere is divided into three regions: north pole region, equatorial region, south pole region, wherein the number of points on the equal latitude ring is equal to 5N side Is an equatorial region, a north pole region above the equatorial region, and a south pole region below the equatorial region;
let i be the line number of the equal latitude ring, the resolution dλ of the point location on the ring along the longitudinal direction is given by the number of points of each line
Let j denote the column number of the point in the corresponding latitude circle, the grid point (i, j) coordinates are calculated as follows:
north pole region:
equatorial region:
region of south pole:
wherein θ and λ represent the latitude and longitude, respectively, of the grid center point on the sphere;
through the formula, the latitude and longitude of the grid center point on the spherical surface under different resolutions can be calculated from the current index value (i, j).
Further, the step 3 includes:
at ico _HEALPix grid divisionUnder the cloth, one-by-one opposite equationThe ' block ' in the coefficient matrix N of (1) is inverted, and the non-block ' parameter is ignored, so as to obtain N -1 Best approximation of (a) represents N * Using a matrix N * Obtaining a spherical harmonic coefficient vector loop iteration solving formula:
combining the formulas (13) to obtain a new expression
Wherein r is (k+1) Is the residual vector of the k +1 iteration,and respectively representing spherical harmonic coefficient vectors obtained by the kth iteration and the kth+1st iteration, wherein U is a normal equation free term vector.
Compared with the prior art, the invention has the beneficial effects that:
the invention provides a grid which is more suitable for ball resonance analysis and improves HEALPix, namely a ico-HEALPix grid. The grid downsampled data not only effectively improves the utilization rate of the data in the spherical harmonic analysis process and the stability of the spherical harmonic analysis structure, but also solves the problem of poor spherical harmonic analysis precision caused by insufficient data sampling in the HEALPix grid high latitude area.
Compared with HEALPix, the ico-HEALPix grid provided by the invention has higher data processing precision in a high-latitude area, and solves the problem of high-latitude data redundancy under the condition of no precision loss compared with the traditional geographic grid.
The invention can realize high-efficiency and high-precision spherical harmonic analysis and provides important technical support and reference for the future research of earth gravity field model construction and universe microwave background radiation.
Drawings
FIG. 1 is a flowchart of an application of a novel spherical uniform distribution grid in spherical harmonic analysis according to an embodiment of the present invention;
FIG. 2 is an expanded view of a ico _HEALPix grid plane in accordance with an embodiment of the present invention; the right graph is obtained by rotating a southern hemisphere of a left graph by 36 degrees, the upper side is k=1, and the lower graph is k=2;
fig. 3 shows a ico _healpix grid spherical distribution of examples k=1 to 4 of the present invention;
FIG. 4 is a block diagonal method matrix generated by a geographic grid and a method matrix corresponding to a ico _HEALPix grid according to an embodiment of the invention;
FIG. 5 is a schematic diagram of an iterative block diagonal least squares method according to an embodiment of the present invention;
FIG. 6 is a diagram of global gravity anomaly data constructed based on a ico _HEALPix grid in accordance with an embodiment of the present invention;
FIG. 7 is a comparison of the order error of the HEALPix grid of the present invention with a modified HEALPix grid spherical harmonic analysis.
Detailed Description
The invention is further illustrated by the following description of specific embodiments in conjunction with the accompanying drawings:
the application of the novel spherical uniform distribution grid in the spherical harmonic analysis is characterized in that firstly, 20 basic grids which are symmetrical in terms of the north-south hemisphere on the spherical surface are generated by combining with a Plastria regular icosahedron structure and being different from 12 basic grids of HEALPix; secondly, continuously subdividing each basic grid to obtain ico _HEALPix grid distribution, and determining a calculation formula of coordinates of the centers of the sub-grids based on the equal area characteristics; and finally, determining the relation between the spherical harmonic analysis order and the grid subdivision hierarchy by using the distribution, performing spherical harmonic analysis, and determining the advantages of ico _HEALPix grid distribution compared with the geographic grid and HEALPix grid in spherical harmonic analysis. The technical flow is shown in figure 1.
The method comprises the following specific steps:
step 1: and (5) determining a basic grid. According to the selection method of the HEALPix basic grids, the 3X 4 total 12 basic grids of the HEALPix are expanded to the 4X 5 total 20 basic grids by combining the excellent characteristics of the icosahedron, and the distribution symmetry of the north-south hemispheres is realized by 36 degrees of rotation of the south-north hemispheres, so that the symmetry can effectively improve the calculation efficiency of the spherical harmonic analysis process.
The method comprises dividing the north to south into 4 rows, uniformly dividing each row into 5 basic grids, wherein each basic grid can be subdivided into N in each direction side ＝2 k Segments and form a sub-grid, k=1, 2,3 …, where N side And k is a parameter representing resolution. For the convenience of calculation, the southern hemisphere of the 4×5 basic grid is integrally rotated by 36 °, and a grid symmetrical about the equator north and south can be obtained, and the plane distribution diagram of the grid after translation is shown in fig. 2. Besides the change of the basic grid distribution, the data special for the HEALPix grid is constructed according to resolution layering, the excellent characteristics of the area and the latitude distribution of the grid and the like are reserved.
Step 2: position coordinates of the sub-grid points are determined. And (3) referring to a sub-grid point position determining method of HEALPix, and combining the equal area and equal latitude distribution characteristics of grids to give spherical coordinates at each grid point under different resolutions (subdivision levels).
Each layer of ico _HEALPix grids is obtained by equally dividing the edges of the previous layer of grids, thus for the resolution parameter N side Or k, each base grid can be divided intoSub-grids, thus the whole sphere is composed ofEqual area but differently shaped sub-grids. As can be seen from FIG. 2, all sub-grid midpoints are located at 5N side -1 equal latitude rings, and each sub-grid has an area of +.>
The number of points on each ring in the polar region is different, only 5 points are on the ring closest to the north pole and the south pole of the sphere, the number of points is gradually increased layer by layer according with an arithmetic progression, and the maximum is 5N side The number of points on the equal latitude ring near the equator is the same and is 5N side A point. Thus, the entire sphere can be divided into three regions: north pole region, equatorial region, south pole region, wherein the number of points on the equal latitude ring is equal to 5N side Is the equatorial region with north (upper) north and south (lower) south. Let i be the line number of the equal latitude ring, the following gives the resolution of the point location on the ring along the longitude direction according to the point number of each line as
Wherein dλ represents the longitudinal resolution;
let j denote the column number of the point in the corresponding latitude circle, the grid point (i, j) coordinates are calculated as follows:
north pole region:
equatorial region:
region of south pole:
through the above formula, the latitude and longitude (θ, λ) of the grid center point on the sphere can be calculated from the current index value (i, j).
The number of grids corresponding to different grid resolutions and the pixel resolutions are shown in table 1.
TABLE 1 number of grids corresponding to different grid resolutions and pixel resolution
The spherical ico _healpix grid distribution in which the subdivision hierarchy k=1 to 4 is shown in fig. 3.
Step 3: the method adopts an iterative block diagonal least square method to realize the spherical harmonic analysis under ico _HEALPix grids, and realizes the recovery of a differential high-order earth gravity field model by utilizing the gravity anomaly under ico _HEALPix grid segmentation.
Spherical harmonic analysis by utilizing gravity anomalies of global geographic grids, wherein the basic observation equation is as follows
Where αg (r, θ, λ) is the spatial gravity anomaly of a point with spherical coordinates (r, θ, λ), r represents the distance from the point (r, θ, λ) to the earth center, θ represents the afterlatitude, λ represents the longitude, GM is the gravitational constant, a is the reference radius, N max The highest order of the gravity field model, n is the order of the gravity field model, m is the order of the gravity field model,representing a completely normalized facial spherical harmonic;
wherein the perfect normalized facial spherical harmonics are expressed as
Wherein the method comprises the steps ofFor normalizing the associated Legendre function, < >>And->The disturbance bit coefficient (namely, the spherical harmonic coefficient) of the spherical harmonic expansion of the gravity field model is obtained. Knowing the gravity anomaly value on the sphere, solving the equation (5) to obtain the spherical harmonic coefficient, and in order to meet the requirement of solving the ultra-high order spherical harmonic coefficient, a block diagonal least square method is generally adopted.
Representing the observation equation (5) as a general vector form
L＝F(X) (8)
Wherein the observed quantity and the spherical harmonic coefficient vector are respectively
Δg t The gravity anomaly value representing the t-th point, the F function is a linear equation about X, and the error equation is described as follows according to a least squares adjustment model of the linear equation system
Wherein v represents the correction vector of observed quantity, A is the coefficient matrix in the adjustment model, the column number is the number of unknowns, namely the total number of the coefficients of the spherical harmonic model, and the line number is the total number of the observed values, which corresponds to the table of the element values
The expression is
If the gravity anomaly covariance matrix is approximately considered to be a diagonal matrix, then the full matrix of gravity anomaly observations
Wherein the method comprises the steps ofRepresenting unit weight variance>Representing the covariance matrix of the observations.
From the error equation, a normal equation can be obtained
Wherein n=a T PA is called coefficient matrix of normal equation, simply called normal matrix, u=a T PL is the normal equation free term vector.
According to the orthogonal characteristic of the spherical harmonic basis function, when the above solution is performed by adopting the gravity anomaly data distributed by the geographic grid, the main diagonal dominant block diagonal form of the method matrix N can be realized by sequencing the to-be-solved quantity X, namely the spherical harmonic coefficients according to a certain rule, as shown in the (left) of figure 4, the values in the middle block range are included,
the outside of the block is a decimal close to 0, so that the equation corresponding to the block one by one can be adopted to solve, and the aim of converting a large matrix into a large number of small matrices is fulfilled.
However, for the gravity anomaly data distribution under the ico _HEALPix grid distribution proposed in this patent, the normal matrix constructed according to the formula (10) does not satisfy the block diagonal form, but the block diagonal dominant matrix, as shown in FIG. 4.
Therefore, the patent proposes an iterative block diagonal least square method for the first time aiming at the step, and can realize an ultra-high order spherical harmonic analysis process under big data, wherein the steps are shown in fig. 5.
From the foregoing, a linear equation is formed between gravity anomaly and spherical harmonic coefficients, whereinThe spherical harmonic comprehensive process can be understood as that the spherical harmonic coefficient obtains the gravity abnormal value through linear transformation, and +.>The spherical harmonic analysis process is adopted, namely, the spherical harmonic coefficient is obtained by linear transformation of the gravity anomaly value.
Under the novel ico _HEALPix grid distribution, the method matrix N is in a non-block diagonal distribution mode, so that the method matrix N is large-scale -1 Is difficult to solve. Because N is a symmetrical positive definite matrix with dominant main diagonal, inverting the ' blocks ' in the normal matrix N one by one and ignoring the non-block ' parameters, N can be obtained -1 Is the best approximation of N * By using the matrix, a cyclic iteration solving formula can be obtained:
wherein r is (k+1) Is the residual vector of the (k+1) th iteration, and the new expression is obtained by combining the above expressions
This expression satisfies the Richardson iterative form of a relaxation factor of 1 when the matrix (I-N * And N) the spectrum radius is smaller than 1, the iteration method is converged, and the final iteration result is consistent and approximates to a true value. By utilizing the flow, the high-precision rapid calculation of the ultra-high order spherical harmonic analysis under the non-geographic grid distribution can be realized.
To verify the effect of the present invention, this patent takes the recovery of the ultra-high order earth gravity field model as an example, and uses the first 3600 order bit coefficients of the gravity field model XGM2019e with the highest current order to simulate and calculate the gravity anomaly values at the midpoints of 20971520 ico _healpix meshes (see fig. 6) with l=10, and the gravity anomaly values at the midpoints of 50331648 HEALPix meshes. Based on the model gravity outlier values, the previous 3599 order bit coefficients of XGM2019e are recovered by using the iterative block diagonal least square method in step 3 of the present patent.
Table 2 recovery of 3600 order spherical harmonic coefficients grid point count comparison statistics using helpix and ico _helpix
According to experimental conditions, for recovering the spherical harmonic coefficient of 3599 order, the ico _HEALPix grid saves the number of points by nearly 1.5 times compared with the HEALPix grid (see table 2), and the precision of the spherical harmonic analysis result is improved by nearly 1 order of magnitude (see fig. 7), so that the application advantage of the ico _HEALPix grid in the spherical harmonic analysis is demonstrated.
In summary, the present invention provides a grid, ico-HEALPix grid, which is a modified HEALPix grid more suitable for spherical harmonic analysis. The grid downsampled data not only effectively improves the utilization rate of the data in the spherical harmonic analysis process and the stability of the spherical harmonic analysis structure, but also solves the problem of poor spherical harmonic analysis precision caused by insufficient data sampling in the HEALPix grid high latitude area. Compared with HEALPix, the ico-HEALPix grid provided by the invention has higher data processing precision in a high-latitude area, and solves the problem of high-latitude data redundancy under the condition of no precision loss compared with the traditional geographic grid. The invention can realize high-efficiency and high-precision spherical harmonic analysis and provides important technical support and reference for the future research of earth gravity field model construction and universe microwave background radiation.
The foregoing is merely illustrative of the preferred embodiments of this invention, and it will be appreciated by those skilled in the art that changes and modifications may be made without departing from the principles of this invention, and it is intended to cover such modifications and changes as fall within the true scope of the invention.
Claims (1)
1. An application of a novel spherical uniform distribution grid in spherical harmonic analysis, which is characterized by comprising the following steps:
step 1: according to a selection method of HEALPix basic grids, generating 20 basic grids which are symmetrical in terms of north-south hemispheres on a sphere by combining with a Plastria regular icosahedron structure, and continuously subdividing each basic grid to obtain a novel spherical uniform distribution grid ico _HEALPix;
the step 1 comprises the following steps:
expanding total 12 basic grids of HEALPix 3×4 to total 20 basic grids of 4×5, rotating the entire southern hemisphere of the basic grid of 4×5 by 36 ° to obtain grids symmetrical about the equator north and south, and subdividing each basic grid into N in each direction side ＝2 k Segments and form a sub-grid, where N side And k is a parameter representing resolution;
the step 1 further includes:
the characteristics of data hierarchical construction, grid equal area and equal latitude distribution of HEALPix grid according to resolution are reserved;
step 2: combining the isosurface integral cloth characteristics of the ico _HEALPix grid, and determining the spherical coordinates at each grid point under different resolutions;
the step 2 comprises the following steps:
the whole sphere is divided into three regions: north pole region, equatorial region, south pole region, wherein the number of points on the equal latitude ring is equal to 5N side Is an equatorial region, a north pole region above the equatorial region, and a south pole region below the equatorial region;
let i be the line number of the equal latitude ring, the resolution dλ of the point location on the ring along the longitudinal direction is given by the number of points of each line
Let j denote the column number of the point in the corresponding latitude circle, the grid point (i, j) coordinates are calculated as follows:
north pole region:
equatorial region:
region of south pole:
wherein θ and λ represent the latitude and longitude, respectively, of the grid center point on the sphere;
through the formula, the latitude and longitude of the grid center point on the spherical surface under different resolutions can be calculated by the current index value (i, j);
step 3: adopting an iterative block diagonal least square method to realize the spherical harmonic analysis under ico _HEALPix grids;
the step 3 comprises the following steps:
under ico _HEALPix grid distribution, one-by-one opposite equationThe ' block ' in the coefficient matrix N of (1) is inverted, and the non-block ' parameter is ignored, so as to obtain N -1 Best approximation of (a) represents N * Using a matrix N * Obtaining a spherical harmonic coefficient vector loop iteration solving formula:
combining the formulas (13) to obtain a new expression
Wherein r is (k+1) Is the residual vector of the k +1 iteration,and respectively representing spherical harmonic coefficient vectors obtained by the kth iteration and the kth+1st iteration, wherein U is a normal equation free term vector.
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Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| AU2002953540A0 (en) * | 2002-12-24 | 2003-01-16 | The University Of Queensland | Correction of non-linear gradients effects on magnetic resonance imaging |
| CN113916197A (en) * | 2021-09-08 | 2022-01-11 | 中国科学院国家天文台 | Celestial sphere observation area estimation method and celestial sphere observation system |
| CN115236759A (en) * | 2022-02-28 | 2022-10-25 | 中国人民解放军战略支援部队信息工程大学 | Hexagonal grid subdivision method for determining earth gravity field |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| AU2002953540A0 (en) * | 2002-12-24 | 2003-01-16 | The University Of Queensland | Correction of non-linear gradients effects on magnetic resonance imaging |
| CN113916197A (en) * | 2021-09-08 | 2022-01-11 | 中国科学院国家天文台 | Celestial sphere observation area estimation method and celestial sphere observation system |
| CN115236759A (en) * | 2022-02-28 | 2022-10-25 | 中国人民解放军战略支援部队信息工程大学 | Hexagonal grid subdivision method for determining earth gravity field |
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