引用本文: 毛政元, 范琳娜, 李霖. 二维空间中距离不确定性的测度方法研究[J]. 武汉大学学报 ( 信息科学版), 2023, 48(12): 1969-1977.
MAO Zhengyuan, FAN Linna, LI Lin. Methodological Research on Measuring Distance Uncertainties in Two-Dimensional Space[J]. Geomatics and Information Science of Wuhan University, 2023, 48(12): 1969-1977.
 Citation: MAO Zhengyuan, FAN Linna, LI Lin. Methodological Research on Measuring Distance Uncertainties in Two-Dimensional Space[J]. Geomatics and Information Science of Wuhan University, 2023, 48(12): 1969-1977.

## Methodological Research on Measuring Distance Uncertainties in Two-Dimensional Space

• 摘要: 距离是空间位置的函数，定量、精确地揭示空间位置不确定性向距离不确定性传递的函数关系具有重要的理论与现实意义，是测绘与地理信息领域亟待解决的重大科学问题。针对该问题现有解决方案的局限性，在满足与不确定点观测位置对应的实际位置在误差圆内服从完全空间随机分布的前提下，推导了二维空间中一个确定点与一个不确定点间以及两个不确定点间距离不确定性的概率分布函数和对应的概率密度函数，并利用后者研究了点位不确定性向距离不确定性传递的规律，为研究与解决距离不确定性问题开辟了新的途径。研究结果表明, 确定点与不确定点间以及两个不确定点间的距离不确定性均服从如下规律：（1）当误差圆半径（对应点位精度）与点间观测距离同时改变时，前者与后者之比与距离不确定性正相关。（2）当误差圆半径保持不变时，距离不确定性与点间观测距离负相关。（3）当点间观测距离保持不变时，距离不确定性与误差圆半径正相关。当误差圆半径与点间观测距离一致时，两个不确定点间距离的不确定性大于确定点和不确定点间距离的不确定性；当该条件不成立时，涉及不确定点数不同的距离不确定性不具可比性。

Abstract:
Objectives Distances are functions of spatial positions. Precisely revealing the functional relationship which quantitatively embodies the transmission of uncertainty from spatial positions to their distance, a key scientific problem in need of being solved urgently in geomatics, has important theoretical and practical significance.
Methods Aiming at the limitation of presently available solution of the above mentioned problem, under the premise of that the real position corresponding with the observed one of an uncertain point follows the complete spatial random distribution within the error circle, we have derived the probability distribution function of the distance uncertainty and the corresponding density function containing an uncertain point and those between two uncertain points respectively in two-dimensional space. The latter has been employed to explore the transmission law of point uncertainties to distance uncertainties, opening up a new way for studying and solving the problem of distance uncertainties.
Results The results show that for all cases: (1) When the radius of the error circle (corresponding to the point position accuracy) and the observed distance between points change simultaneously, their ratio has a significant positive correlation with the level of distance uncertainties. (2) When the former remains constant, the distance uncertainty has a significant negative correlation with the latter. (3) When the latter remains constant, the distance uncertainty has a significant positive correlation with the former.
Conclusions As far as the distance uncertainty of cases containing an uncertain point and the one of those between two uncertain points are concerned, the latter is obviously greater than the former when the radius of the error circle and the observed distance between points are consistent for both of them. Otherwise they are not comparable.

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