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An Improved QTM Subdivision Model with Approximate Equal-area
ZHAO Xuesheng, YUAN Zhengyi , ZHAO Longfei, ZHU Sikun
College of Geoscience and Surveying Engineering, China University of Mining & Technology(Beijing), Beijing 100083, China
First author: ZHAO Xuesheng (1967—), male, professor, majors in modeling & 3D visualization of the global discrete grids. E-mail： zxs@cumtb.edu.cn
Corresponding author:E-mail： yuanzhengyi001@163.com
Abstract: To overcome the defect of large area deformation in the traditional QTM subdivision model, an improved subdivision model is proposed which based on the “parallel method” and the thought of the equal area subdivision with changed-longitude-latitude. By adjusting the position of the parallel, this model ensures that the grid area between two adjacent parallels combined with no variation, so as to control area variation and variation accumulation of the QTM grid. The experimental results show that this improved model not only remains some advantages of the traditional QTM model(such as the simple calculation and the clear corresponding relationship with longitude/latitude grid, etc), but also has the following advantages: ①this improved model has a better convergence than the traditional one. The ratio of area_max/min finally converges to 1.38, far less than 1.73 of the “parallel method”; ②the grid units in middle and low latitude regions have small area variations and successive distributions; meanwhile, with the increase of subdivision level, the grid units with large variations gradually concentrate to the poles; ③the area variation of grid unit will not cumulate with the increasing of subdivision level.
Key words: discrete global grid system     QTM     parallel ring method     geometry deformation of grid
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1 “纬线环法”剖分方案基本原理 1.1 “分割纬线”的纬度计算方法

 图 1“纬线环”及其在XOY面上的投影 Fig. 1 “Parallel loop” and its projection on the XOY

 图 2 分割纬线求解过程示意图 Fig. 2 Schematic diagram of solving segmentation parallel

(1) 首先确定第1条分割纬线的位置(i=1)。此时，B0=90°、r1=RcosB0r2=RcosB1n表示剖分层次。代入式(4)可得

(2) 根据第1条分割纬线的位置确定第2条的位置(i=2)。将r1=RcosB1r2=RcosB2代入式(4)可得

(3) 以此类推，递归求解各分割纬线的位置，第i条纬线的纬度Bi见式(9)，其中0≤i≤2nB0=90°

(4) 接下来按“平分经度”的方法确定模型的经度间隔，得到剖分单元节点的经纬度坐标。借鉴“纬线法”QTM剖分方案[20]，用大圆弧线连接三角形格网的左右两边，而底边用两点间的纬线代替。

1.2 剖分单元面积计算

 图 3 剖分单元面积的近似计算 Fig. 3 Approximate computation of area of subdivision unit

A、B、C为球面上任意3点，X1X2X3分别为A、B、C 3点的向量(笛卡儿坐标系坐标)，则球面三角形ABC的曲面面积为

2 试验结果及分析

 图 4 两种剖分方案及其叠加效果 Fig. 4 Two kinds of subdivision schemes and their superposition
2.1 剖分模型收敛性分析

 层次 三角形个数 纬线法 改进方法 area-max/min area_SD area-max/min area_SD 1 4 1.353 812 565 0.517 136 445 1.186 661 755 0.471 404 521 2 16 1.530 463 278 0.343 590 689 1.269 627 924 0.304 228 748 3 64 1.677 029 977 0.258 487 516 1.338 029 710 0.167 748 119 4 256 1.714 716 465 0.236 651 898 1.364 597 114 0.095 451 834 5 1024 1.724 207 897 0.229 680 661 1.371 282 896 0.054 390 527 6 4096 1.726 584 916 0.227 455 224 1.372 956 182 0.030 400 488 7 16 384 1.727 178 871 0.226 769 805 1.373 372 874 0.016 687 041 8 65 536 1.727 326 077 0.226 565 927 1.373 472 871 0.009 031 593 9 262 144 1.727 359 000 0.226 507 913 1.373 485 470 0.004 836 264 10 1 048 576 1.727 344 400 0.226 498 479 1.373 420 285 0.002 568 582

 图 5area_max/min和area_SD随剖分层次的收敛特征 Fig. 5 Convergent characters of area_max/min and area_SD by partition levels increasing
2.2 剖分单元面积变化分布区段研究

 % 层次 -∞~-20% -20%~-10% -10%~-5% -5%~5% 5%~10% 10%~20% 20%~+∞ 1 0.00 50.00 0.00 0.00 25.00 25.00 0.00 2 12.50 12.50 0.00 43.75 12.50 0.00 18.75 3 12.50 21.88 0.00 28.13 14.06 15.63 7.81 4 6.25 22.66 10.16 22.27 12.50 19.53 6.64 5 3.13 23.05 14.84 21.00 10.55 22.27 5.18 6 4.64 22.85 12.70 21.95 10.69 21.80 5.37 7 3.87 24.28 11.43 22.36 10.75 21.20 6.12 8 3.49 24.46 11.93 22.17 10.68 20.68 6.59 9 3.68 24.31 11.74 22.33 10.70 20.57 6.67 10 3.77 24.29 11.74 22.25 10.69 20.57 6.69

 % 层次 -∞~-5% -5%~-3% -3%~-1% -1%~1% 1%~3% 3%~5% 5%~+∞ 1 50.00 0.00 0.00 25.00 0.00 0.00 25.00 2 31.25 0.00 6.25 31.25 0.00 0.00 31.25 3 4.69 20.32 17.19 18.75 12.50 9.38 17.19 4 2.73 1.95 28.51 32.80 29.29 1.56 3.13 5 0.88 0.88 16.41 62.30 17.97 0.59 0.98 层次 -∞~-0.75% -0.75%~-0.5% -0.5%~-0.25% -0.25%~0.25% 0.25%~0.5% 0.5%~0.75% 0.75%~+∞ 6 6.35 12.55 15.06 31.90 15.04 12.60 6.54 7 1.99 1.46 15.41 62.10 15.58 1.47 2.02 8 0.62 0.64 2.22 93.02 2.22 0.64 0.63 9 0.17 0.20 0.90 97.50 0.89 0.20 0.17 10 0.04 0.05 0.26 99.30 0.26 0.05 0.04
2.3 剖分单元面积变化位置分布研究

 图 6 剖分单元面积变化率的位置分布(第5层) Fig. 6 Location distributions of the area variation rate of subdivision units (the 5th level)

 图 7 剖分单元面积变化率的位置分布(第6层) Fig. 7 Location distributions of the area variation rate of subdivision units (the 6th level)

 图 8 剖分单元面积变化率的位置分布(第7层) Fig. 8 Location distributions of the area variation rate of subdivision units (the 7th level)

3 结 语

(1) 面积变化小。随着格网的不断细化，格网单元面积的最大最小值之比越来越大，标准差越来越小，变化速度越来越小，最终都趋于收敛。相比之下，改进后的“纬线环法”剖分模型的面积收敛性更好，这表明改进模型的格网面积分布更加均匀，各格网单元之间有更好的相似性。改进后的剖分模型，格网单元的面积变化率更小，至第10层剖分，“纬线法”QTM模型只有22%左右的格网面积变化率在5%以内，而“纬线环法”QTM模型99%以上的格网面积变化率被控制在0.25%以内。

(2) 位置分布明确。随着剖分层次的增加，改进后的“纬线环法”剖分模型，变化边界向两极移动，中低纬度区域近似等面积剖分。此外，面积变化不会随剖分层次的增加而积累，相反会逐渐减小。这一特点，非常有利于误差控制与分析。

(3) 计算简便。相比于“投影法”以及“小圆弧法”得到的等面积球面离散格网模型，该改进后的QTM剖分模型由于不需要复杂的数学变换，计算更加简便。

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http://dx.doi.org/10.11947/j.AGCS.2016.20140598

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文章信息

ZHAO Xuesheng, YUAN Zhengyi, ZHAO Longfei, ZHU Sikun

An Improved QTM Subdivision Model with Approximate Equal-area

Acta Geodaeticaet Cartographica Sinica, 2016, 45(1): 112-118.
http://dx.doi.org/10.11947/j.AGCS.2016.20140598