﻿ 近区地形直接与间接影响的棱柱模型算法
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1. 地理信息工程国家重点实验室, 陕西 西安 710054;
2. 信息工程大学地理空间信息学院, 河南 郑州 450001;
3. 西安测绘研究所, 陕西 西安 710054

Prism Algorithms for the Near-zone Direct and Indirect Topographic Effects
MA Jian1,2,3 , WEI Ziqing1,3
1. State Key Laboratory of Geo-information Engineering, Xi'an 710054, China;
2. Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China;
3. Xi'an Research Institute of Surveying and Mapping, Xi'an 710054, China
Foundation support: The National Natural Science Foundation of China (Nos. 41674025; 41674082); The Open Research Foundation of State Key Laboratory of Geo-information Engineering (Nos. SKLGIE2016-M-1-5; SKLGIE2016-M-3-2)
First author: MA Jian(1988—), female, PhD, majors in physical geodesy. E-mail:majian_19881006@163.com
Abstract: It is necessary to calculate the direct effect on gravity and the indirect effects on the geoid and quasi-geoid when reducing the topography based on Helmert's second condensation method for solving boundary value problems. The traditional integration algorithms for the near-zone direct and indirect topographic effects arein the double integral form and give some approximate errors when taking the integral kernel at the center of the grid as the average of the grid integral kernel. Furthermore, the traditional integral algorithms for the direct and indirect effects have singularity in the innermost area, which increases the computational complexity. In this paper, the prism formulae of the near-zone direct and indirect effects are deduced, which improves the calculation accuracy on the one hand and simplifies the computation process by removing the singularity in the innermost area on the other hand. To overcome the planar approximation error of the prism models, the curvature of the earth can be taken into account. Finally, experimental results show that the prism algorithm for the topographic effect considering the earth's curvature is recommended when a high-accuracy geoid/quasi-geoid is required.
Key words: near-zone direct effect     near-zone indirect effect     Helmert's second condensation method     innermost-zone singularity     prism algorithms     considering the earth's curvature

1 地形影响的传统积分公式

(1)

(2)

(3)

(4)

(5)

(6)

2 近区地形影响的棱柱模型算法推导

 图 1 地形网格与计算点相对位置示意图 Fig. 1 The schematic diagram of a topographic grid away from the computing point

2.1 直接影响

(7)

(8)

(9)

(10)

(11)

(12)

f(x)为变量x的函数，g(y)为变量y的函数，根据式(13)

(13)

(14)
2.2 间接影响

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

f(θ)=sin θ，上下界θ2=π/2、θ1=0，则式(25)左端等于1/2，右端等于π/4，显然是不成立的。当θ1θ2差值减小时，式(25)两端的差异(误差)也将随之缩小。需要特别说明的是，当fkθ/cos θ(k为常数，此时积分核与自变量为线性关系)时，式(25)是恒成立的。上述对一元积分式(25)的分析同样适用于二重积分式(24)。

3 试验

 地形影响 算法 最小值 最大值 平均值 RMS 直接影响/(mGal) 0.5°×0.5° -64.680 61.066 -2.069 10.767 1°×1° -63.962 64.690 -0.672 10.548 3°×3° -63.452 66.896 0.439 10.545 间接影响/m 0.5°×0.5° 0.008 0.751 0.152 0.186 1°×1° 0.007 0.756 0.151 0.186 3°×3° 0.001 0.741 0.141 0.177

 mGal 算法 算法 最小差值 最大差值 平均差值 RMS 未顾及曲率的棱柱算法 0.5°×0.5° 0.001 0.229 0.047 0.059 1°×1° 0.001 0.237 0.048 0.061 3°×3° 0.001 0.241 0.049 0.062 积分算法 0.5°×0.5° -10.660 9.713 -2.813 3.362 1°×1° -10.663 9.709 -2.814 3.364 3°×3° -10.662 9.710 -2.814 3.364

 cm 算法 算法 最小差值 最大差值 平均差值 RMS 未顾及曲率的棱柱算法 0.5°×0.5° -0.241 -0.012 -0.070 0.082 1°×1° -0.445 -0.031 -0.145 0.168 3°×3° -1.146 -0.157 -0.499 0.556 积分算法 0.5°×0.5° -3.399 0.045 -0.703 0.899 1°×1° -2.992 0.132 -0.553 0.740 3°×3° -1.717 1.245 0.154 0.405

 地形影响 分辨率 最小值 最大值 平均值 RMS 直接影响/(mGal) 30″×30″ -54.342 90.365 8.240 14.518 3″×3″ -53.128 92.032 10.244 15.694 间接影响/m 30″×30″ 0.007 0.764 0.151 0.187 3″×3″ 0.007 0.766 0.151 0.187

 地形影响 分辨率 最小差值 最大差值 平均差值 RMS 直接影响/(mGal) 30″×30″ -0.568 5.304 3.837 4.040 3″×3″ -0.158 -0.015 -0.106 0.110 间接影响/cm 30″×30″ -0.187 0.686 0.048 0.161 3″×3″ -0.964 -0.060 -0.291 0.341

4 结论

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http://dx.doi.org/10.11947/j.AGCS.2018.20170369

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#### 文章信息

MA Jian, WEI Ziqing

Prism Algorithms for the Near-zone Direct and Indirect Topographic Effects

Acta Geodaetica et Cartographica Sinica, 2018, 47(11): 1429-1436
http://dx.doi.org/10.11947/j.AGCS.2018.20170369