Construction of quality evaluation indicator system for diamond discrete global grid systems

Figures & data

Figure 1. Grids with different cell shape: (a) diamond; (b) triangle; (c) hexagon; (d) square.

Figure 2. Flowchart of the quality evaluation indicator model. The abbreviations in Discrete Global Grid Systems are described in Table 1. The abbreviations among the Quantitative Indicators are defined in Appendix 1. DATA refers to an observation data matrix.

Figure 3. Regular polyhedrons: (a) octahedron; (b) spherical octahedron; (c) icosahedron.

Figure 4. Diamond grids with different aperture: (a) aperture-4 diamond; (b) aperture-9 diamond.

Table 1. Discrete global grid systems used this paper.

Shape	Diamond	Grids	Abbreviation
Regular polyhedron	Octahedron	Icosahedral Snyder Equal Area aperture 4 Diamond	ISEA4D
	Icosahedron	Octahedral Recursive Division aperture 4 Diamond	OR4D
Projection	Snyder	Octahedral non-Recursive Division aperture 4 Diamond	OnonR4D
	Recursive	Icosahedral Recursive Division aperture 4 Diamond	IR4D
	Non-Recursive	Icosahedral non-Recursive Division aperture 4 Diamond	InonR4D

Figure 5. Spherical polygon P. Where O is the center of the sphere, p₁, p₂, p₃, p₄ are vertices of the spherical polygon P.

Figure 6. Polygon P and its dihedral angle. p₁, p₂, p₃, p₄ are the vertices of spherical polygon P; α₁, α₂, α₃ and α₄ are the inner angles of the spherical polygon P; ${\overrightarrow{\mathit{n}}}_{\mathit{O}p23}$ is the normal vector of plane Op₂p₃.

Figure 7. Spherical distance. Where O is the center of the sphere; r is the radius of the sphere; p_i and p_j are points on the sphere; θ_ij is the angle between ${\overrightarrow{Op}}_i$ and ${\overrightarrow{Op}}_j$; d_ij is the spherical distance between p_i and p_j.

Figure 8. Distance between two adjacent grid cell centers. Where q_P, q_P1, q_P2, q_P3 and q_P4 is the center of the diamond grid cell P, P₁, P₂, P₃ and P₄.

Figure 9. Parameters for CSD. Where q_i and q_j are the centers of grid cell P_i and P_j, respectively; p_ij is the shared edge between grid cell P_i and P_j; c_pij is the midpoint of p_ij; q_ij is the line connecting grid centers q_i and q_j; c_qij is the midpoint of q_ij; CSD is the geodesic distance between c_pij and c_qij.

Table 2. Statistics of indicators of OnonR4D-5 model (R = 6371.007 km).

Indicator	Mean	Standard deviation
A	124527.824 km^2	40592.787
D	376.626 km	62.105
CD	374.607 km	63.705
CSD	6.735 km	8.067
ZSC	0.818	0.032
FRAC	0.626	0.006
SI	0.818	0.032

Table 3. Correlation coefficient matrix of the OnonR4D-5 model.

	A	D	CD	CSD	ZSC	FRAC	SI
A	1.000
D	0.978	1.000
CD	0.979	0.995	1.000
CSD	−0.427	−0.384	−0.464	1.000
ZSC	0.210	0.029	0.078	−0.352	1.000
FRAC	−0.786	−0.679	−0.714	0.517	−0.747	1.000
SI	0.211	0.030	0.079	−0.352	1.000	−0.748	1.000

Table 4. Eigenvalue and cumulative contribution rate of variance for the OnonR4D-5 model.

Components	Eigenvalue	Contribution rate of variance (%)	Cumulative contribution rate of variance (%)
1	4.201	60.013	60.013
2	2.110	30.139	90.152
3	0.669	9.562	99.714
4	0.019	0.267	99.981
5	0.001	0.015	99.996
6	1.089 × 10^−5	0.004	100
7	7.990 × 10^−9	1.141 × 10^−7	100

Figure 10. Curve of eigenvalues of the OnonR4D-5 model.

Table 5. Influence coefficient of the principal components of the OnonR4D-5 model.

Components	A	D	CD	CSD	ZSC	FRAC	SI
1	0.971	0.991	0.976	−0.266	0.023	−0.716	0.024
2	0.169	−0.010	0.030	−0.226	0.991	−0.666	0.991
3	−0.138	−0.131	−0.211	0.937	−0.130	0.190	−0.130

Figure 11. Path diagrams of the OnonR4D-5 Model. The green and red arrow indicate the positive and negative, respectively, and the number on each of arrows represent the absolute value of the influence coefficients of the principal components of the OnonR4D-5 model (similarly hereinafter).

Table 6. Correlation matrix of the OR4D-5 model.

	A	D	CD	CSD	ZSC	FRAC	SI
A	1.000
D	0.870	1.000
CD	0.858	0.964	1.000
CSD	−0.128	0.007	−0.178	1.000
ZSC	0.213	−0.294	−0.245	−0.277	1.000
FRAC	−0.554	−0.076	−0.116	0.291	−0.930	1.000
SI	0.213	−0.294	−0.244	−0.277	1.000	−0.930	1.000

Table 7. Eigenvalue and cumulative contribution rate of variance for the OR4D-5 model.

Components	Eigenvalue	Contribution rate of variance (%)	Cumulative contribution rate of variance (%)
1	3.182	45.455	45.455
2	2.892	41.316	86.771
3	0.900	12.864	99.635
4	0.024	0.339	99.974
5	0.002	0.023	99.997
6	0.000	0.003	100.000
7	3.180 × 10^−9	4.543 × 10^−8	100.000

Table 8. Influence coefficients of the principal components of the OR4D-5 model.

Components	A	D	CD	CSD	ZSC	FRAC	SI
1	0.939	0.983	0.971	−0.062	−0.129	−0.243	−0.128
2	0.338	−0.165	−0.140	−0.179	0.985	−0.964	0.985
3	−0.007	0.041	−0.148	0.982	−0.112	0.105	−0.112

Figure 12. Path diagram of the OR4D-5 model.

Table 9. Correlation matrix of the InonR4D-5 model.

	A	D	CD	CSD	ZSC	FRAC	SI
A	1.000
D	0.981	1.000
CD	0.981	0.992	1.000
CSD	−0.460	−0.442	−0.541	1.000
ZSC	0.097	−0.092	−0.044	−0.127	1.000
FRAC	−0.817	−0.698	−0.727	0.436	−0.647	1.000
SI	0.098	−0.091	−0.043	−0.127	1.000	−0.648	1.000

Table 10. Eigenvalue and cumulative contribution rate of variance for the InonR4D-5 model.

Components	Eigenvalue	Contribution rate of variance (%)	Cumulative contribution rate of variance (%)
1	4.012	57.310	57.310
2	2.279	32.563	89.873
3	0.703	10.046	99.918
4	0.004	0.053	99.971
5	0.002	0.028	99.999
6	7.413 × 10^−5	0.001	100.000
7	6.279 × 10^−8	8.970 × 10^−7	100.000

Figure 13. Path diagrams of the InonR4D-5 model.

Table 11. Influence coefficients of the principal components of the InonR4D-5 model.

Components	A	D	CD	CSD	ZSC	FRAC	SI
1	0.980	0.983	0.962	−0.305	−0.014	−0.740	−0.013
2	0.103	−0.085	−0.042	−0.09	0.998	−0.651	0.998
3	−0.16	−0.158	−0.265	0.945	−0.044	0.16	−0.044

Table 12. Correlation matrix of the IR4D-5 model.

	A	D	CD	CSD	ZSC	FRAC	SI
A	1.000
D	0.833	1.000
CD	0.810	0.961	1.000
CSD	−0.205	−0.150	−0.339	1.000
ZSC	0.107	−0.461	−0.428	−0.063	1.000
FRAC	−0.467	0.101	0.080	0.135	−0.929	1.000
SI	0.107	−0.461	−0.428	−0.063	1.000	−0.929	1.000

Table 13. Eigenvalue and cumulative contribution rate of the variance for the IR4D-5 model.

Components	Eigenvalue	Contribution rate of variance (%)	Cumulative contribution rate of variance (%)
1	3.431	49.018	49.018
2	2.625	37.500	86.518
3	0.918	13.119	99.636
4	0.025	0.361	99.997
5	0.000	0.002	99.999
6	6.186 × 10^−5	0.001	100.000
7	1.560 × 10^−8	2.229 × 10^−7	100.000

Table 14. Influence coefficients of the principal components of the IR4D-5 model.

Components	A	D	CD	CSD	ZSC	FRAC	SI
1	0.95	0.96	0.922	−0.144	−0.202	−0.173	−0.202
2	0.304	−0.274	−0.254	−0.066	0.978	−0.983	0.978
3	−0.049	−0.03	−0.226	0.985	−0.029	0.046	−0.029

Figure 14. Path diagrams of the IR4D-5 model.

Table 15. Statistics of indicators for the ISEA4D-5 model (R = 6371.007 km).

Indicator	Mean	Standard Deviation
A	49811.716 km^2	0
D	241.173 km	7.048
CD	240.599 km	7.391
CSD	2.052 km	3.953
ZSC	0.821	0.024
FRAC	0.282	0.004
SI	0.821	0.024

Figure 15. Curve of eigenvalues for the ISEA4D-5 model.

Table 16. Correlation matrix of the ISEA4D-5 model.

	D	CD	CSD	ZSC	FRAC	SI
D	1.000
CD	0.894	1.000
CSD	−0.074	−0.363	1.000
ZSC	−1.000	−0.894	0.073	1.000
FRAC	−1.000	−0.894	0.074	1.000	1.000
SI	−1.000	−0.894	0.073	1.000	1.000	1.000

Table 17. Eigenvalue and cumulative contribution rate of the variance for the ISEA4D-5 model.

Components	Eigenvalue	Contribution rate of variance (%)	Cumulative contribution rate of variance (%)
1	4.856	80.925	80.925
2	1.058	17.634	98.559
3	0.086	1.435	99.994
4	0.000	0.006	100
5	2.169 × 10^−7	3.615 × 10^−6	100
6	2.242 × 10^−8	3.737 × 10^−7	100

Table 18. Influence coefficients of the principal components for the ISEA4D-5 model.

Components	D	CD	CSD	ZSC	FRAC	SI
1	−0.998	−0.903	0.046	0.998	0.998	0.998
2	−0.023	−0.345	0.995	0.023	0.023	0.023

Figure 16. Path diagrams of the ISEA4D-5 model.

Table A1. The Goodchild criteria.

No.	Criterion	Simplification (symbol)
1	Grids cover the Earth’s surface without overlapping	–
2	The grids have equal area	Area (A)
3	The grids have the consistent topology	–
4	The grids have identical shape	Shape Index (SI)
5	The grids are compact	Compactness (ZSC)
6	The grids’ edges are straight by projection	Edge (D)
7	The midpoint of shared edge between adjacent grids coincides with the midpoint of great arc connecting the two grids’ centers	Cell wall midpoint ratio (CSD)
8	The grid model has hierarchy	–
9	Each grid has unique reference point	–
10	Distances between any adjacent grids are equal;	Centers’ Distance (CD)
11	Reference points and grids have regularities, and they are corresponding to a coding system	Fractal Dimension (FRAC)
12	There is an easy transformation between the grid system and geographic coordinate system	–
13	The grid model contains grids with any arbitrary resolution	–
14	The grids’ centers coincide maximally with their reference points	–

Table A2. Influence coefficients of the principal components of the OnonR4D-6 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.976	0.995	0.987	−0.191	0.028	−0.709	0.029
2	0.164	−0.016	0.003	−0.147	0.995	−0.684	0.995
3	−0.105	−0.101	−0.161	0.970	−0.092	0.143	−0.092

Table A3. Influence coefficients of the principal components of the OnonR4D-7 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.980	0.997	0.993	−0.137	0.039	−0.753	0.039
2	0.156	−0.023	−0.014	−0.099	0.997	−0.641	0.997
3	−0.076	−0.074	−0.118	0.986	−0.064	0.104	−0.064

Table A4. Influence coefficients of the principal components of the OnonR4D-8 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.982	0.998	0.996	−0.098	0.050	−0.797	0.050
2	0.147	−0.033	−0.028	−0.067	0.997	−0.588	0.997
3	−0.054	−0.053	−0.085	0.993	−0.045	0.075	−0.045

Table A5. Influence coefficients of the principal components of the OR4D-6 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.941	0.985	0.980	−0.039	−0.144	−0.270	−0.144
2	0.336	−0.164	−0.152	−0.118	0.987	−0.960	0.987
3	−0.006	0.026	−0.091	0.992	−0.077	0.072	−0.077

Table A6. Influence coefficients of the principal components of the OR4D-7 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.941	0.986	0.984	−0.028	−0.154	−0.309	−0.154
2	0.337	−0.161	−0.156	−0.082	0.987	−0.949	0.987
3	−0.006	0.016	−0.061	0.996	−0.054	0.051	−0.054

Table A7. Influence coefficients of the principal components of the OR4D-8 model.

component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.940	0.987	0.986	−0.027	−0.160	−0.360	−0.160
2	0.339	−0.159	−0.156	−0.060	0.986	−0.932	0.986
3	−0.010	0.006	−0.047	0.998	−0.040	0.040	−0.040

Table A8. Influence coefficients of the principal components of the InonR4D-6 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.987	0.987	0.976	−0.224	−0.007	−0.788	−0.007
2	0.093	−0.095	−0.074	−0.06	0.998	−0.6	0.998
3	−0.121	−0.12	−0.2	0.97	−0.032	0.125	−0.032

Table A9. Influence coefficients of the principal components of the InonR4D-7 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.991	0.99	0.984	−0.162	0.004	−0.834	0.004
2	0.083	−0.105	−0.094	−0.039	0.999	−0.54	0.999
3	−0.088	−0.088	−0.146	0.984	−0.023	0.094	−0.023

Table A10. Influence coefficients of the principal components of the InonR4D-8 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.993	0.991	0.988	−0.116	0.013	−0.876	0.013
2	0.074	−0.114	−0.108	−0.026	0.999	−0.473	0.999
3	−0.063	−0.063	−0.104	0.99	−0.016	0.069	−0.016

Table A11. Influence coefficients of the principal components of the IR4D-6 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.947	0.956	0.861	−0.198	−0.173	−0.158	−0.172
2	0.308	−0.284	−0.237	−0.099	0.984	−0.985	0.984
3	−0.074	−0.05	−0.378	0.973	−0.034	0.058	−0.034

Table A12. Influence coefficients of the principal components of the IR4D-7 model.

Component	A	D	CD	CSD	ZSC	SI	FRAC
1	0.95	0.963	0.946	−0.1	−0.22	−0.206	−0.219
2	0.306	−0.265	−0.256	−0.043	0.975	−0.977	0.975
3	−0.035	−0.022	−0.147	0.992	−0.02	0.034	−0.02

Table A13. Influence coefficients of the principal components of the IR4D-8 model.

Component	A	D	CD	CSD	ZSC	FRAC	SI
1	0.948	0.965	0.959	−0.071	−0.233	−0.252	−0.233
2	0.313	−0.255	−0.251	−0.026	0.971	−0.967	0.971
3	−0.025	−0.017	−0.101	0.995	−0.011	0.023	−0.011

Table A14. Influence coefficients of the principal components of the ISEA4D-6 model.

Component	D	CD	CSD	ZSC	FRAC	SI
1	−0.999	−0.960	0.018	0.999	0.999	0.999
2	0.017	−0.205	0.999	−0.017	−0.017	−0.017

Table A15. Influence coefficients of the principal components of the ISEA4D-7 model.

Component	D	CD	CSD	ZSC	FRAC	SI
1	−0.999	−0.982	−0.001	0.999	0.999	0.999
2	0.027	−0.127	1	−0.028	−0.028	−0.028

Table A16. Influence coefficients of the principal components of the ISEA4D-8 model.

Component	D	CD	CSD	ZSC	FRAC	SI
1	−0.999	−0.992	−0.005	0.999	0.999	0.999
2	0.027	−0.082	1	−0.026	−0.026	−0.026

Figure A1. Path diagrams of the OnonR4D Model: (a) OnonR4D-6; (b) OnonR4D-7; (c) OnonR4D-8. The green and red arrow indicate the positive and negative, respectively, and the number on each of arrows represent the absolute value of the influence coefficients of the principal components of the OnonR4D-6/7/8 model (similarly hereinafter).

Figure A2. Path diagrams of the OR4D Model: (a) OR4D-6; (b) OR4D-7; (c) OR4D-8.

Figure A3. Path diagrams of the InonR4D Model: (a) InonR4D-6; (b) InonR4D-7; (c) InonR4D-8.

Figure A4. Path diagrams of the IR4D Model: (a) IR4D-6; (b) IR4D-7; (c) IR4D-8.

Figure A5. Path diagrams of the ISEA4D Model: (a) ISEA4D-6; (b) ISEA4D-7; (c) ISEA4D-8.

Data availability statement

All grids in this study are generated by the open-source DGGS implementation – DGGRID, which can be retrieved from https://www.discreteglobalgrids.org/software/. And the original data for five grids can be obtained from https://github.com/joelfl/Data-for-Quality-Evaluation-Indicator-System-.git.