誤差理論與測(cè)量平差基礎(chǔ)教學(xué)-第九講ppt課件

1、第二章第二章 參參 數(shù)數(shù) 平平 差差Parametric Least-Squares AdjustmentParametric Least-Squares Adjustment補(bǔ)充知識(shí)補(bǔ)充知識(shí)1 1 多元函數(shù)極值多元函數(shù)極值 設(shè)函數(shù)設(shè)函數(shù) 求求F F的極值點(diǎn)，即求以下方程組的的極值點(diǎn)，即求以下方程組的解解第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment上述方程的上

2、述方程的解解 叫做駐點(diǎn)，叫做駐點(diǎn)，對(duì)于實(shí)踐問對(duì)于實(shí)踐問題駐點(diǎn)就是題駐點(diǎn)就是極值點(diǎn)。極值點(diǎn)。),(21txxxFF00021txFxFxF000021txFxFxF),(21txxx補(bǔ)充知識(shí)補(bǔ)充知識(shí)2 2 矩陣的導(dǎo)數(shù)矩陣的導(dǎo)數(shù) 設(shè)矩陣設(shè)矩陣Y Y中的元素都是變量中的元素都是變量x x的可導(dǎo)函數(shù)的可導(dǎo)函數(shù)第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustmentmnmmnnyy

3、yyyyyyyY212222111211dxdydxdydxdydxdydxdydxdydxdydxdydxdydxdYmnmmnn212222111211 右式即是矩陣右式即是矩陣Y Y對(duì)變量對(duì)變量x x的導(dǎo)數(shù)的導(dǎo)數(shù)補(bǔ)充知識(shí)補(bǔ)充知識(shí)2 2 矩陣的導(dǎo)數(shù)矩陣的導(dǎo)數(shù) 性質(zhì)性質(zhì)第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS AdjustmentTTdxdYdxdY)(0dxdCdxdYdx

4、Yd )(dxdZdxdYdxZYd)(ZdxdYdxdZYdxYZd)(1122335544補(bǔ)充知識(shí)補(bǔ)充知識(shí)3 3 列向量對(duì)列向量的導(dǎo)數(shù)列向量對(duì)列向量的導(dǎo)數(shù) 設(shè)列向量設(shè)列向量Y Y中的元素都是列向量中的元素都是列向量X X中元素的函數(shù)中元素的函數(shù)第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment右兩式即是列向量右兩式即是列向量Y Y對(duì)列向量對(duì)列向量X X的導(dǎo)數(shù)的導(dǎo)

5、數(shù)nmxxxXyyyY2121 , nmmmnnxyxyxyxyxyxyxyxyxydXdY212221212111)(21nxYxYxYdXdY補(bǔ)充知識(shí)補(bǔ)充知識(shí)3 3 列向量對(duì)列向量的導(dǎo)數(shù)列向量對(duì)列向量的導(dǎo)數(shù)第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment 性質(zhì)性質(zhì)11223355440dXdCdXdYdXYd)(dXdZdXdYdXZYd)(dXdYCdXCY

6、d)(dXdYZdXdZYXdZYdTTnmmT111)(dXdVPVdXPVVdTT2)(66復(fù)習(xí)復(fù)習(xí)第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment平差：在多余觀測(cè)的根底上，根據(jù)一定的觀測(cè)模型，按照平差：在多余觀測(cè)的根底上，根據(jù)一定的觀測(cè)模型，按照一定的數(shù)學(xué)原那么，對(duì)觀測(cè)結(jié)果進(jìn)展合理的調(diào)整，求得一一定的數(shù)學(xué)原那么，對(duì)觀測(cè)結(jié)果進(jìn)展合理的調(diào)整，求得一組沒有矛盾的最

7、可靠結(jié)果，并評(píng)定精度，這種處置方法和組沒有矛盾的最可靠結(jié)果，并評(píng)定精度，這種處置方法和過程稱為丈量平差。過程稱為丈量平差。最小二乘原理最小二乘原理minPVVT最小二乘平差計(jì)算本質(zhì)上是一個(gè)極值問題。最小二乘平差計(jì)算本質(zhì)上是一個(gè)極值問題。V觀測(cè)值的平差矯正數(shù)向量觀測(cè)值的平差矯正數(shù)向量minpvv1 1、Introduction of Parametric Least-Squares Introduction of Parametric Least-Squares AdjustmentAdjustment2 2、Mathematics Model of Parametric Least-Mathe

8、matics Model of Parametric Least-Squares AdjustmentSquares Adjustment3 3、Normal EquationNormal Equation第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment4 4、Steps of Parametric Least-Squares AdjustmentSteps of

9、 Parametric Least-Squares Adjustment5 5、Some ExamplesSome Examples1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric

10、LS AdjustmentCABQuestionQuestion： 要確定平面三角形的外形，等精度觀要確定平面三角形的外形，等精度觀測(cè)三個(gè)內(nèi)角，求各角的最小二乘估值測(cè)三個(gè)內(nèi)角，求各角的最小二乘估值 這樣一個(gè)問題中，未知數(shù)個(gè)數(shù)為幾？獨(dú)立未知數(shù)這樣一個(gè)問題中，未知數(shù)個(gè)數(shù)為幾？獨(dú)立未知數(shù)個(gè)數(shù)為幾？個(gè)數(shù)為幾？1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction a

11、nd Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS AdjustmentL3L1L2X1X2CAB1 1Perfect ConditionPerfect Condition11 xL22 Lx180213xxLqt unknowns, n equations t unknowns, n equations qIf no errors, the If no errors, the equations is compatible equations is compatible

12、 qBut errors exist, equations is incompatible. But errors exist, equations is incompatible. 1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametri

13、c LS AdjustmentParametric LS Adjustment2 2If we have the true errorsIf we have the true errors111 xL222 xL1802133xxLqBut we cannot obtain the true errors ,the X is But we cannot obtain the true errors ,the X is also not obtained also not obtained 1 1、Introduction of Parametric Least-Squares Adjustme

14、ntIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment3 3Regard X and Regard X and as unknowns,append the as unknowns,append the additional condition additional

15、 condition 111 xvL222 xvL1802133xxvLqThe additional condition is LS principleThe additional condition is LS principle假設(shè)我有一千萬，我就能買一棟房子。 我有一千萬嗎？沒有。 所以我依然沒有房子。 假設(shè)我有翅膀，我就能飛。 我有翅膀嗎？沒有。 所以我也沒方法飛。 假設(shè)把整個(gè)太平洋的水倒出， 也澆不熄我對(duì)他愛情的火 。 整個(gè)太平洋的水全部倒得出嗎？不能。 所以，我并不愛他。 1 1、Introduction of Parametric Least-Squares Adjustmen

16、tIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment4 4The Error Equations The Error Equations 111 Lxv222 Lxv3123180vxxL qVResidual errorVResidual error，殘差，殘差q

17、 Unknowns Unknowns，未知參數(shù)，未知參數(shù)q Adjustment Values Adjustment Values，平差值，平差值VLLX1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS Adjustment

18、Parametric LS Adjustment4 4The Error Equations The Error Equations 從數(shù)學(xué)角度上來看，最小二乘問題就是求殘差二從數(shù)學(xué)角度上來看，最小二乘問題就是求殘差二次型次型VTPVVTPV的極小值點(diǎn)問題的極小值點(diǎn)問題2321222211)180()()(LxxLxLxPVVT1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The

19、Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment4 4The Error Equations The Error Equations 更為普通的，設(shè)更為普通的，設(shè)nvvvV21txxxX21nLLLL21ndddD21nnntbatbatbaA222111 于是于是LDXAV 誤差方程誤差方程1 1、Introduction of Parametric Least-Squares AdjustmentIntroducti

20、on of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment5 5Examples Examples 例211 設(shè)有平面三角形，如圖（211） ， 為了確定三角形的形狀等精度觀測(cè)了三個(gè)內(nèi)角，其觀測(cè)值分別為1L、2L、3L，求三個(gè)角度的最或然值。 ABC1x2x）圖（1125 5Examples

21、Examples 例212 如圖（212）所示的水準(zhǔn)網(wǎng)，A 為已知水準(zhǔn)點(diǎn)，1P、2P為未知水準(zhǔn)點(diǎn)，觀測(cè)高差為1h、2h、3h（其中箭頭指向高程增加方向） ，試求1P、2P點(diǎn)的高程。 1PA2P1h2h3h圖（212） AAHxvhxxvhHxvh23312221113232212111hHxvhxxvhHxvAA or or5 5Examples Examples or or 例 1 2 3 設(shè) 從 三 個(gè) 已 知 點(diǎn)1P、2P、3P，對(duì) 未 知 點(diǎn)P進(jìn) 行 距 離 測(cè)量 ， 觀 測(cè) 值 為321SSS、。 現(xiàn) 由 三 個(gè)已 知 點(diǎn) 的 坐 標(biāo)),(111yxP、),(222yxP、),(33

22、3yxP， 求P點(diǎn) 的 坐 標(biāo)),(yx。 ),(333yxP),(111yxP),(222yxP),(yxP1S2S3S）圖（212232333232222212111)()()()()()(yyxxvSyyxxvSyyxxvS323233223222121211)()()()()()(SyyxxvSyyxxvSyyxxv1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The In

23、troduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment參數(shù)平差的概念參數(shù)平差的概念 首先根據(jù)詳細(xì)問題選取一組獨(dú)立且足夠的首先根據(jù)詳細(xì)問題選取一組獨(dú)立且足夠的未知數(shù)作參數(shù)，列出這些未知參數(shù)與全部觀未知數(shù)作參數(shù)，列出這些未知參數(shù)與全部觀丈量之間的關(guān)系式，即誤差方程；然后根據(jù)丈量之間的關(guān)系式，即誤差方程；然后根據(jù)這些關(guān)系式按最小二乘原理，求出各未知參這些關(guān)系式按最小二乘原理，求出各未知參數(shù)及觀丈量的最或然值，并精度估計(jì)。數(shù)及觀丈量的最或

24、然值，并精度估計(jì)。1 1、Introduction of Parametric Least-Squares AdjustmentIntroduction of Parametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment6 6NoteNoteq只需存在多余觀測(cè)才會(huì)存在平差問題。由于僅有只需存在多余觀測(cè)才會(huì)存

25、在平差問題。由于僅有必需觀測(cè)時(shí)，觀測(cè)方程有獨(dú)一解。普通必需觀測(cè)時(shí)，觀測(cè)方程有獨(dú)一解。普通n n表示觀表示觀測(cè)值個(gè)數(shù)，測(cè)值個(gè)數(shù)，t t是必需觀測(cè)數(shù)，是必需觀測(cè)數(shù)，n n大于大于t t。 q參數(shù)平差的未知數(shù)個(gè)數(shù)必需等于必需觀測(cè)個(gè)數(shù)。參數(shù)平差的未知數(shù)個(gè)數(shù)必需等于必需觀測(cè)個(gè)數(shù)。q未知參數(shù)必需函數(shù)獨(dú)立，即未知參數(shù)間不存在條未知參數(shù)必需函數(shù)獨(dú)立，即未知參數(shù)間不存在條件關(guān)系。件關(guān)系。q未知參數(shù)可以是間接觀丈量，所以參數(shù)平差也叫未知參數(shù)可以是間接觀丈量，所以參數(shù)平差也叫間接平差間接平差 。q誤差方程可以是非線性方式。誤差方程可以是非線性方式。 2 2、Mathematics Model of Parametr

26、ic Least-Squares Mathematics Model of Parametric Least-Squares AdjustmentAdjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment11函數(shù)模型函數(shù)模型對(duì)一個(gè)丈量的實(shí)踐問題，假設(shè)有對(duì)一個(gè)丈量的實(shí)踐問題，假設(shè)有n n個(gè)觀測(cè)個(gè)觀測(cè)值，值，t t個(gè)未知參數(shù)，對(duì)應(yīng)的存在個(gè)未知參數(shù)，對(duì)應(yīng)的存在

27、n n個(gè)矯正數(shù)個(gè)矯正數(shù)2 2、Mathematics Model of Parametric Least-Squares Mathematics Model of Parametric Least-Squares AdjustmentAdjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS AdjustmentnvvvV21txxxX21nLLLL21ndddD21nnnt

28、batbatbaA222111LAXDLDXAV觀測(cè)方程觀測(cè)方程誤差方程誤差方程11函數(shù)模型函數(shù)模型2 2、Mathematics Model of Parametric Least-Squares Mathematics Model of Parametric Least-Squares AdjustmentAdjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adj

29、ustment? Weight matrix of l (l=D-L)? expectation of VLDXAV lXAVl=D-L11函數(shù)模型函數(shù)模型2 2、Mathematics Model of Parametric Least-Squares Mathematics Model of Parametric Least-Squares AdjustmentAdjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS

30、 AdjustmentParametric LS Adjustment對(duì)于觀測(cè)值向量對(duì)于觀測(cè)值向量L L的真誤差向量的真誤差向量 lAX真11函數(shù)模型函數(shù)模型兩邊取數(shù)學(xué)期望兩邊取數(shù)學(xué)期望0)( )()()E(lEAXlEAXE真真2 2、Mathematics Model of Parametric Least-Squares Mathematics Model of Parametric Least-Squares AdjustmentAdjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introd

31、uction and Principle of Parametric LS AdjustmentParametric LS Adjustment22隨機(jī)模型隨機(jī)模型0)(E0120P觀測(cè)向量的真誤差的期望為零觀測(cè)向量的真誤差的期望為零權(quán)矩陣正定權(quán)矩陣正定2212222111221nnnnn3 3、Normal EquationNormal Equation法方程法方程第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS Adjustme

32、ntParametric LS Adjustment1 1The Principle of The Least Squares Estimation The Principle of The Least Squares Estimation q可求得獨(dú)一估值可求得獨(dú)一估值最小PVVT lXAV聯(lián)立聯(lián)立0120P根據(jù)根據(jù)3 3、Normal Equation Normal Equation 法方程法方程第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Param

33、etric LS AdjustmentParametric LS Adjustment2 2The Normal Equation The Normal Equation 極值方程極值方程0)(TTXdPVVdXddVPVXdPVVdTT2)(由誤差方程得由誤差方程得AXddV極值方程為極值方程為0PVAT誤差方程代入上式：誤差方程代入上式：0PlAXPAATTNPAATUPlAT0UXN3 3、Normal Equation Normal Equation 法方程法方程第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The

34、 Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment2 2The Normal Equation The Normal Equation qNnormal equation coefficient matrix(Nnormal equation coefficient matrix(系數(shù)陣系數(shù)陣qUnormal equation vectorUnormal equation vector自在項(xiàng)自在項(xiàng)0UXN0PVATor參數(shù)的解向量：參數(shù)的解向量：UNX13 3、Normal Equati

35、onNormal Equation法方程法方程第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS AdjustmentNote: Note: 1 1法方程系數(shù)陣法方程系數(shù)陣NN的維數(shù)的維數(shù)t tt t2 2法方程的秩法方程的秩r(N)=r(ATPA)=r(A)=tr(N)=r(ATPA)=r(A)=t3 3當(dāng)當(dāng)r(A)=tr(A)=t時(shí)，稱矩陣時(shí)，稱矩陣A A列滿秩，為保證列滿秩，為保證

36、A A列滿秩，未知參數(shù)選列滿秩，未知參數(shù)選 取是關(guān)鍵取是關(guān)鍵4 4只需只需NN滿秩時(shí)，法方程才有獨(dú)一解滿秩時(shí)，法方程才有獨(dú)一解3 3、Normal EquationNormal Equation法方程法方程第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment3 3 is the least point of VTPV is the least point of VTP

37、V設(shè)另有解設(shè)另有解 ，矯正數(shù)向量，矯正數(shù)向量V1V1，只需，只需PVVPVVTT11命題就得證命題就得證)(2)()( )()(1111111VVPVVVPVVPVVVVVPVVVPVVPVVTTTTTT由于由于lXAVlXAV11)(11XXAVVX1X3 3、Normal EquationNormal Equation第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustm

38、ent3 3 is the least point of VTPV is the least point of VTPV4 4 is the unbiased estimate is the unbiased estimate 0)()(1111VVPVVPVVPVVTTTXXUNX1真真XPAXANlPEANPlAENUENXETTT1111)( )()()(5 5 The Expectation of V is .? The Expectation of V is .?4 4、 Steps of Parametric Least-Squares Adjustment Steps of Pa

39、rametric Least-Squares Adjustment第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS Adjustment1 1選取足夠且獨(dú)立的參數(shù)作為未知量選取足夠且獨(dú)立的參數(shù)作為未知量(t (t個(gè)個(gè)) )； 2 2列出誤差方程式列出誤差方程式(n(n個(gè)個(gè)) )； 3 3依誤差方程式組成法方程；依誤差方程式組成法方程； 4 4解法方程求得未知向量估值；解法方程求得未知向

40、量估值； 5 5回代回代X X向量，求出矯正數(shù)和觀測(cè)值平差值；向量，求出矯正數(shù)和觀測(cè)值平差值； 6 6精度估計(jì)。精度估計(jì)。)(X)0(PlAXPAATT)(lXAV)(lXAV)(VLL5 5、Special Example:Special Example:第九講第九講 參數(shù)平差概述與原理參數(shù)平差概述與原理The Introduction and Principle of The Introduction and Principle of Parametric LS AdjustmentParametric LS AdjustmentThe Normal Equation for Independent Obs.The Normal Equation for Independent Obs.法方程的普通方式法方程的普通方式0PlAXPAATT觀測(cè)值獨(dú)立觀測(cè)值獨(dú)立, ,權(quán)矩陣為對(duì)角陣權(quán)矩陣為對(duì)角陣0PVAT0 , , 0 , 0ptvpbvpav( (此組公式可作為平差檢核公式此組公式可作為平差檢核公式02121212121ptvpbvpavvvvppptttbbbaaannnnn5 5、Sp