测量误差理论的真值中心论和测得值中心论

叶晓明; 丁士俊; 师会生

doi:10.3969/j.issn.2096-9015.2021.03.04

测量误差理论的真值中心论和测得值中心论

doi: 10.3969/j.issn.2096-9015.2021.03.04

1.
武汉大学测绘学院，武汉 430079
2.
中国航空工业集团公司长城计量测试技术研究所，北京 100095

详细信息

作者简介:
叶晓明（1963-），武汉大学测绘学院副教授，研究方向：测量仪器及误差理论，邮箱：xmye@sgg.whu.edu.cn

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出版历程
- 网络出版日期: 2021-04-13

True-Value Centered Theory and Measured-Value Centered Theory in Measurement Error Theories

1.
School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China
2.
AVIC Changcheng Institute of Metrology & Measurement, Beijing 100074, China

摘要

摘要: 传统经典测量误差理论认为测得值（观测值）是随机变量，真值是常量；但新概念测量理论认为测得值（观测值）是常量，真值是随机变量。本文通过概率论概念的回顾，对二种不同的常量和随机变量概念解释进行对比分析，指出传统测量理论对基本数学概念的曲解以及由此导致的诸多概念逻辑困境，并阐明新概念测量理论的基本概念逻辑和误差评价原理。
- 测量 /
- 测量误差 /
- 随机变量 /
- 方差 /
- 不确定度 /
- 概率论
Abstract: The classical measurement error theory postulates that the measured (observed) value is a random variable while the true value is a constant. In contrast, a new-concept measurement theory postulates that the measured value is a constant while the true value is a random variable. After reviewing concepts in the probability theory, this paper makes a comparative analysis of the above-mentioned theories in respect of the different interpretations of the constant and the random variable. It is pointed out that the traditional measurement theory misinterprets basic mathematical concepts, leading to conceptual or logical trouble. The basic concepts and logic and principles of error evaluation are explained in the new measurement theory.
- measurement /
- measurement error /
- random variable /
- variance /
- uncertainty /
- probability theory

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图 1 传统测量误差理论的概念示意图

Figure 1. Conceptual diagram of the traditional measurement error theory

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图 2 新概念理论的概念原理示意图

Figure 2. Schematic diagram of concepts in the new theory

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图 3 测距仪周期误差的规律性和随机性

Figure 3. Regularity and randomness in the periodic error of a rangefinder

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图 4 四舍五入误差的规律性和随机性

Figure 4. Regularity and randomness in the rounding error

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图 5 直流电压测量中交流电干扰误差的规律性和随机性

Figure 5. Regularity and randomness in the AC interference error in DC voltage measurements

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图 6 电子噪声误差的规律性和随机性

Figure 6. Regularity and randomness in an electronic noise error

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图 7 三段基线

Figure 7. Three-section baselines

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表 1 概率论教科书对离散型随机变量概念的描述

Table 1. The concept of a discrete random variable in the probability theory

$ X $	$ {x}_{1} $	$ {x}_{2} $	…	$ {x}_{i} $	…
$ P $	$ {p}_{1} $	$ {p}_{2} $	…	$ {p}_{i} $	…

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表 2 公式（10）（11）（12）中的概念类别比较

Table 2. Comparison of concept categories in Formulas (10), (11), and (12)

	观测值${q_k}$	观测值${q_j}$	测得值$\bar q $
公式（10）	常数	——	常数
公式（11）	随机变量	常数	常数
公式（12）	随机变量	——	随机变量

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表 3 传统测量误差理论中的数学期望缺失问题

Table 3. Lack of mathematical expectation in the traditional measurement error theory

测得值$ x $	随机误差$ r=x-E\left(x\right) $	系统误差$ s=E\left(x\right)-{x}_{T} $	真值$ {x}_{T} $
$\left\{ {\begin{array}{*{20}{l}} {E(x) = ???} \\ {{u^2}(x) = E{{[x - E(x)]}^2}} \end{array} } \right.$	$\left\{ {\begin{array}{*{20}{l}} {E(r) = 0} \\ {{u^2}(r) = {u^2}(x)} \end{array} } \right.$	$\left\{ {\begin{array}{*{20}{l}} {E(s) = ???} \\ {{u^2}(s) = 0} \end{array} } \right.$	$\left\{ {\begin{array}{*{20}{l}} {E({x_T}) = ???} \\ {{u^2}({x_T}) = 0} \end{array} } \right.$

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表 4 新概念测量误差理论中的概念解释

Table 4. Concepts in the new measurement error theory

测得值$ {x}_{0} $	误差$ \varDelta $	真值$ {X}_{T} $
$\left\{ {\begin{array}{*{20}{l}} {E({x_0}) = {x_0}} \\ {{u^2}({x_0}) = 0} \end{array} } \right.$	$\left\{ {\begin{array}{*{20}{l}} {E(\varDelta ) = 0} \\ {{u^2}(\varDelta ) = E({\varDelta ^2})} \end{array} } \right.$	$\left\{ {\begin{array}{*{20}{l}} {E({X_T}) = {x_0}} \\ {{u^2}({X_T}) = {u^2}(\varDelta )} \end{array} } \right.$

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表 5 三段基线的全组合观测值

Table 5. Combination of all observations of three-section baselines

线段	AB	BC	CD	AC	BD	AD
观测值	$ {x}_{1} $	$ {x}_{2} $	$ {x}_{3} $	$ {x}_{4} $	$ {x}_{5} $	$ {x}_{6} $

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