1、第一章第一章 误差理论与最小二乘原理误差理论与最小二乘原理Error Theory and The Least Squares PrincipleError Theory and The Least Squares Principle第一讲第一讲 绪绪 论（论（Introduction）(复习复习)思考题思考题 1 1、什么是多余观测，它的作用是什么？、什么是多余观测，它的作用是什么？2 2、什么是测量条件，它与误差有什么关系？什么是测量条件，它与误差有什么关系？3 3、什么是平差？平差能否消除误差？、什么是平差？平差能否消除误差？4 4、最小二乘原理的数学表达式是什么、最小二乘原理的数学表达
2、式是什么?有几种特例？有几种特例？No.2 Classes of Errors and Characteristics2 2、All observations contain errorsAll observations contain errors上节课内容回顾：上节课内容回顾：3 3、Necessary and redundant observationNecessary and redundant observation4 4、The role of redundant observationThe role of redundant observation1 1、Observation
3、s generally require some form Observations generally require some form instrumentationinstrumentation that is used by an that is used by an observerobserver in in certain certain environmentenvironmentNo.2 Classes of Errors and Characteristics上节课内容回顾：上节课内容回顾：5 5、Adjustment Adjustment aimsaims to eli
4、minate to eliminate inconsistenciesinconsistencies among the observationsamong the observations6 6、Adjustment Adjustment followsfollows the Least Squares the Least Squares 第二讲第二讲 误差分类及其特性误差分类及其特性Classes of Errors and Characteristics1、True Value and True Error2、The Classes of Errors3、The probability
5、Characteristics of Random Error4、The Statistics Meaning of True ValueNo.2 Classes of Errors and Characteristics1 The Condition1 The Conditions s of Surveying of Surveying1 1、True Value and True ErrorTrue Value and True ErrorObserverObserverInstrumentInstrumentEnvironmentEnvironmentObjectiveObjective
6、No.2 Classes of Errors and Characteristics1 1、True Value and True ErrorTrue Value and True Error2 Equal Precision Surveying2 Equal Precision SurveyingObserving in the same conditionObserving in the same conditionQuestion:Question:what is the same condition?No.2 Classes of Errors and Characteristics1
7、 1、True Value and True ErrorTrue Value and True Error3 True Value and Estimate Value3 True Value and Estimate ValueTrue True ValueValue（真值）（真值）:反映一个量真正大小、绝对准确的数值反映一个量真正大小、绝对准确的数值以一定准确程度反演一个量大小的数值以一定准确程度反演一个量大小的数值Estimate Estimate ValueValue（估值）（估值）:No.2 Classes of Errors and Characteristics1 1、True
8、Value and True ErrorTrue Value and True Error3 True Value and Estimate Value3 True Value and Estimate ValueNoteNote：1 1、真值一般无法求得、真值一般无法求得2 2、一个量观测值和平差值是真值还是估值？、一个量观测值和平差值是真值还是估值？No.2 Classes of Errors and Characteristics1 1、True Value and True ErrorTrue Value and True Error4 True 4 True ErrorError（真
9、误真误差）差）True Error is the difference between an observation of a quantity and true valueOROR True Error is the difference between the estimate value of a quantity and true valueNo.2 Classes of Errors and Characteristics1 1、True Value and True ErrorTrue Value and True Error4 True 4 True ErrorError（真误真
10、误差）差）For:True value：X ；Estimate value:L；True error:thenXLNo.2 Classes of Errors and Characteristics1 1、True Value and True ErrorTrue Value and True Error4 True Error4 True ErrorNote:Note:一般情况下，在测量中虽然真值是客观存在的，一般情况下，在测量中虽然真值是客观存在的，但观测值的真值及真误差都是不可能知道的，而观但观测值的真值及真误差都是不可能知道的，而观测值的某些函数值的真值却是可以知道。根据观测测值的某些
11、函数值的真值却是可以知道。根据观测值的函数值与其真值的差异，可以判断观测值中是值的函数值与其真值的差异，可以判断观测值中是否存在粗差。否存在粗差。(Since X can never be known,will never be known as well)No.2 Classes of Errors and Characteristics4)4)、True ErrorTrue ErrorExample1:Example1:设，平面三角形内角观测值为设，平面三角形内角观测值为L L1 1，L L2 2，L L3 3 三角形三角形内角和内角和的真误差的真误差闭合差（闭合差（Close Error
12、Close Error）W=(L1+L2+L3)W=(L1+L2+L3)估估-180-1800 0 闭合差闭合差的真误差的真误差WW真真=(=(L1+L2+L3)L1+L2+L3)真真-180-1800 000闭合差的真闭合差的真值值WW=WWWW真真WW闭合差的真误差闭合差的真误差No.2 Classes of Errors and Characteristics4)4)、True ErrorTrue ErrorExample2:Example2:设，某量真值为设，某量真值为X X，两观测值为两观测值为L L1 1，L L2 2 双次观测之差定义为：双次观测之差定义为：d=Ld=L1 1-L
13、-L2 2双次观测之差的真值为：双次观测之差的真值为：d d真真=L L1 1真真L L2 2真真X XX X0 0 双次观测之差的真误差双次观测之差的真误差 d dd d d d真真dd较差较差No.2 Classes of Errors and Characteristics1)1)、Blunders or Gross Blunders or Gross ErrorsErrors（粗差）（粗差）2 2、The Classes of ErrorsThe Classes of Errors2)2)、Systematic Systematic ErrorsErrors（系统误差）（系统误差）3)
14、3)、Random Random ErrorsErrors（偶然误差）（偶然误差）No.2 Classes of Errors and Characteristics1)1)、Blunders or Gross ErrorsBlunders or Gross Errors2 2、The Classes of ErrorsThe Classes of Errors Blunders are mistakes or gross errors.It may be Blunders are mistakes or gross errors.It may be caused by the wrong r
15、eading of instruments,caused by the wrong reading of instruments,transposing numbers or equipment failures.transposing numbers or equipment failures.BlunderBlunderare usually large and easily detectedare usually large and easily detected.No.2 Classes of Errors and Characteristics1)1)、Blunders or Gro
16、ss ErrorsBlunders or Gross Errors2 2、The Classes of ErrorsThe Classes of Errors Characteristics:Characteristics:magnitude is significantly magnitude is significantly large/small/different in comparison large/small/different in comparison to the measured values(abnormal to the measured values(abnorma
17、l observation)observation)No.2 Classes of Errors and Characteristics1)1)、Blunders or Gross ErrorsBlunders or Gross Errors2 2、The Classes of ErrorsThe Classes of Errors Source:Source:Carelessness of the observer;Carelessness of the observer;Failure of instrument;Failure of instrument;Abnormity of env
18、ironment.Abnormity of environment.观测时大数读错观测时大数读错 ；计算机输入数据错误；计算机输入数据错误；航测相片判读错误；航测相片判读错误；控制网起始数据错误。控制网起始数据错误。No.2 Classes of Errors and Characteristics1)1)、Blunders or Gross ErrorsBlunders or Gross Errors2 2、The Classes of ErrorsThe Classes of Errors Effect:Effect:Inhomogeneous observable;Inhomogeneo
19、us observable;Treatment:Treatment:Carefulness;Carefulness;Checking through close error Checking through close error；DetectingDetecting No.2 Classes of Errors and Characteristics1)1)、Blunders or Gross ErrorsBlunders or Gross Errors2 2、The Classes of ErrorsThe Classes of Errors Example:Example:Measuri
20、ng a distance Measuring a distance 31.3m,31.1m,31.2m,13.1m,31.1m31.3m,31.1m,31.2m,13.1m,31.1m 注意：经典平差中，要求粗差消灭在平差前，注意：经典平差中，要求粗差消灭在平差前，今后我们一般认为，待平差的观测值无粗差！今后我们一般认为，待平差的观测值无粗差！No.2 Classes of Errors and Characteristics2)2)、Systematic ErrorsSystematic Errors2 2、The Classes of ErrorsThe Classes of Error
21、sSystematic error affect the observations in Systematic error affect the observations in thethesame way,hence they are hard to detect by same way,hence they are hard to detect by repeated observation.repeated observation.钢钢尺量距中的尺尺量距中的尺长误长误差差.测距仪的加常数和乘常数测距仪的加常数和乘常数.形形变测变测量中的周期量中的周期误误差差.天文天文测测量中的人差量中的
22、人差.大大气气折光差折光差No.2 Classes of Errors and Characteristics2)2)、Systematic ErrorsSystematic Errors2 2、The Classes of ErrorsThe Classes of ErrorsCharacteristics:.Same sign and value;.Same sign and value;.Constant error or bias;Constant error or bias;.Systematic varieties;.Systematic varieties;.Accumulati
23、ve.AccumulativeNo.2 Classes of Errors and Characteristics2)2)、Systematic ErrorsSystematic Errors2 2、The Classes of ErrorsThe Classes of ErrorsTreatment:.Must be detected and corrected;.Must be detected and corrected;.Using certain procedures during measurement;Using certain procedures during measure
24、ment;.Eliminate it in calculatingEliminate it in calculating;No.2 Classes of Errors and Characteristics2)2)、Systematic ErrorsSystematic Errors2 2、The Classes of ErrorsThe Classes of ErrorsExample:level surveyNo.2 Classes of Errors and Characteristics3)3)、Random ErrorsRandom Errors2 2、The Classes of
25、ErrorsThe Classes of ErrorsIt results from accidental and unknown It results from accidental and unknown combination of causes beyond the combination of causes beyond the con-con-troltrol of the observer.of the observer.测测角角误误差照准差照准误误差差读数误读数误差外界差外界条条 件件变变化化仪仪器器误误差差 No.2 Classes of Errors and Charact
26、eristics3)3)、Random ErrorsRandom Errors2 2、The Classes of ErrorsThe Classes of ErrorsCharacteristics:Characteristics:.From some random reasons.From some random reasons .Results from all kinds of reasons.Results from all kinds of reasons.Single random error is no law.Single random error is no law.The
27、 whole of random errors obeys.The whole of random errors obeys statisticalstatistical law law No.2 Classes of Errors and Characteristics3)3)、Random ErrorsRandom Errors2 2、The Classes of ErrorsThe Classes of ErrorsTreatment:Treatment:Cannot be generally eliminated;Cannot be generally eliminated;Howev
28、er,it can be minimized by However,it can be minimized by taking redundant observations and taking redundant observations and applying the so called“Method of applying the so called“Method of Least Squares”.Least Squares”.No.2 Classes of Errors and Characteristics3)3)、Random ErrorsRandom Errors2 2、Th
29、e Classes of ErrorsThe Classes of ErrorsNote:Note:偶然误差是在执行测量规范，消除了粗偶然误差是在执行测量规范，消除了粗差和系统误差后仍然残存的小误差。差和系统误差后仍然残存的小误差。在一切测量中，偶然误差是不可避免在一切测量中，偶然误差是不可避免的，经典平差就是认为观测值中仅含的，经典平差就是认为观测值中仅含有偶然误差有偶然误差 No.2 Classes of Errors and Characteristics2 2、The Classes of ErrorsThe Classes of ErrorsGross errorGross erro
30、rSystematic errorsSystematic errorsConclusions:Conclusions:系统误差和偶然误差同时存在，理想情况是平差系统误差和偶然误差同时存在，理想情况是平差前尽量消除或减弱系统误差的影响，使偶然误差前尽量消除或减弱系统误差的影响，使偶然误差居主要成分；居主要成分；系统误差和偶然误差是相对的，在一定条件下，系统误差和偶然误差是相对的，在一定条件下，可相互转化；可相互转化；即便存在系统误差仍可进行平差即便存在系统误差仍可进行平差，但平差结果，但平差结果不会理想，精度指标也是虚假的；不会理想，精度指标也是虚假的；今后的学习中今后的学习中，我们总假设观测值
31、仅含偶然误差；，我们总假设观测值仅含偶然误差；平差理论的新发展平差理论的新发展，目前也包括了处理粗差和系，目前也包括了处理粗差和系统误差的分支。统误差的分支。No.2 Classes of Errors and CharacteristicsAssumption:Assumption:3 Probability Characteristics of Random Error All measurements are free of gross errors and All measurements are free of gross errors and corrected for all s
32、ystematic errorscorrected for all systematic errorsExample:Example:No.2 Classes of Errors and CharacteristicsExample:Example:triangular networkWWWWWWWWWWWWWW3 Probability Characteristics of Random Errorinnnid nni d 误差的误差的区间区间为负值为负值为正值为正值总数总数个数个数频率频率个数个数频率频率0.000.500.501.001.001.501.502.002.002.502.5
33、03.003.003.503.50 1219078513915900.150.110.100.060.050.020.010.000.300.220.200.120.100.040.020.00123104755527201000.150.130.090.070.030.020.010.000.300.260.180.140.060.040.020.002441941531066635190和和4030.504140.50817inNo.2 Classes of Errors and Characteristics规律规律:1.1.闭合差在数值上不会超过一定的界限；闭合差在数值上不会超过一定的
34、界限；2.2.绝对值小的闭合差比大闭合差的个数多；绝对值小的闭合差比大闭合差的个数多；3.3.绝对值相等的正负闭合差个数大致相等。绝对值相等的正负闭合差个数大致相等。3 Probability Characteristics of Random ErrorNo.2 Classes of Errors and CharacteristicsExample:Example:0.10.10.20.30.20.1o0.20.10.3d 1.1.以误差区间为底，以误差以误差区间为底，以误差出现在该区间的频率除出现在该区间的频率除以区间间隔的商为高以区间间隔的商为高2.2.长方形的面积表示长方形的面积表示
35、误差出现的频率，误差出现的频率，其和等于其和等于1 13.3.长方形的高表示误差出现的密长方形的高表示误差出现的密度度4.4.当当n n趋近于无穷大时，各频率趋近于无穷大时，各频率趋近于一个完全确定的值，趋近于一个完全确定的值，这个值就是误差出现的概率这个值就是误差出现的概率5.5.一定的测量条件对应一定的测量条件对应一定的误差分布一定的误差分布No.2 Classes of Errors and CharacteristicsExample:Example:0.30.2 o1.00.10.20.30.20.12.01.03.05.00.3（区间概率）pn,)(,0fddi连续光滑的曲线线小矩
36、形上端的阶梯形折（概率密度）（概率密度）No.2 Classes of Errors and CharacteristicsExample:Example:0.30.2 o1.00.10.20.30.20.12.01.03.05.00.3 0)(fNo.2 Classes of Errors and Characteristics从理论上分析：从理论上分析：中心极限定理中心极限定理 若随机变量若随机变量 是众多随机变量之和，即是众多随机变量之和，即 如果如果 相互独立，且对相互独立，且对 的影响均匀的小，则当的影响均匀的小，则当 相当大时，随机变量趋于服从正态分布。相当大时，随机变量趋于服从正
37、态分布。ynxxxy21ixyn偶然误差服从正态分布偶然误差服从正态分布n21偶 Review the normal distribution),(2aNX正态分布：正态分布：正态分布的密度函数：正态分布的密度函数：)(exp(21)(22axxf2,XDaXE数学期望和方差：数学期望和方差：今后，我们将正态分布作为研究偶然误差的数学工具。今后，我们将正态分布作为研究偶然误差的数学工具。21x)(xfNo.2 Classes of Errors and Characteristics1在一定的测量条件下，偶然误差的数值不超过一在一定的测量条件下，偶然误差的数值不超过一定的限值，或者说超出一定限
38、值的偶然误差出现的定的限值，或者说超出一定限值的偶然误差出现的概率为零。概率为零。2绝对值小的误差比绝对值大的误差出现的概率大。绝对值小的误差比绝对值大的误差出现的概率大。3绝对值相等的正负误差出现的概率相同。绝对值相等的正负误差出现的概率相同。（Limitation）（centrality）（symmetry）Characteristics:No.2 Classes of Errors and CharacteristicsNotes:Notes:1.1.界限性是讨论极限误差的理论依据；界限性是讨论极限误差的理论依据；2.2.聚中性表明，偶然误差愈接近零，其分布愈密；聚中性表明，偶然误差愈接
39、近零，其分布愈密；3.3.对称性表明，偶然误差有相互抵消的性质对称性表明，偶然误差有相互抵消的性质;0dfE0lim)(nEin 22221LELeLf 22221ef4.4.观测值观测值L L与其真误差的分布密度函数。与其真误差的分布密度函数。No.2 Classes of Errors and Characteristics2)2)、Definition of True ValueDefinition of True Value4、The Statistics Meaning of True ValueExpectation of the quantity only contains Ex
40、pectation of the quantity only contains random errors.random errors.1)1)、It is a Theoretical ValueIt is a Theoretical ValueNo.2 Classes of Errors and Characteristics2)2)、Definition of True ValueDefinition of True ValueTrue value:X=L+True value:X=L+,then,then E(X)=E(L)+E(E(X)=E(L)+E()Hence,E(Hence,E(
41、)=0We can say X=E(L)4、The Statistics Meaning of True ValueNo.2 Classes of Errors and Characteristics3)3)、L and L and have the same distributionhave the same distribution1-1-2=(L1-E(L1)-(L2-E(L2)=L1-L22=(L1-E(L1)-(L2-E(L2)=L1-L2,22221LELeLf 22221ef4、The Statistics Meaning of True Value 第二讲第二讲 误差分类及其特
42、性误差分类及其特性Classes of Errors and Characteristics1 1、测量误差可分为三类；测量误差可分为三类；ConclusionsConclusions：2 2、偶然误差在统计意义上具有三个特性；偶然误差在统计意义上具有三个特性；3 3、真值客观存在，但无法得到；真值客观存在，但无法得到；4 4、真值在统计学意义上是总体平均；真值在统计学意义上是总体平均；5 5、观测值和对应真误差同分布；观测值和对应真误差同分布；6 6、偶然误差的数学期望等于零。偶然误差的数学期望等于零。例题例题 1 1、在下列情况下用钢尺量距，使测量结果含有误、在下列情况下用钢尺量距，使测量结果含有误 差，判别误差性质。差，判别误差性质。（1 1）尺长不准确；）尺长不准确；（2 2）尺长检定过程中，尺长与标准尺长比）尺长检定过程中，尺长与标准尺长比 较产生的误差；较产生的误差；（3 3）尺子不水平；）尺子不水平；（4 4）估读小数不准确。）估读小数不准确。例题例题 2 2、根据误差分布曲线判定哪一组观测值精度高。、根据误差分布曲线判定哪一组观测值精度高。)(f212211)(f)(f例题例题 3 3、说明误差分布曲线出现下面情况的原因。、说明误差分布曲线出现下面情况的原因。)(f
