1、第二章 參 數(shù) 平 差,Parametric Least-Squares Adjustment,補(bǔ)充知識(shí),1 多元函數(shù)極值,設(shè)函數(shù) 求F的極值點(diǎn)，即求下列方程組的解,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,上述方程的 解 叫做駐點(diǎn)， 對(duì)于實(shí)際問(wèn) 題駐點(diǎn)就是 極值點(diǎn)。,補(bǔ)充知識(shí),2 矩陣的導(dǎo)數(shù),設(shè)矩陣Y中的元素都是變量x的可導(dǎo)函數(shù),第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,右式即是矩陣Y對(duì)

2、變量x的導(dǎo)數(shù),補(bǔ)充知識(shí),2 矩陣的導(dǎo)數(shù),性質(zhì),第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,1,2,3,5,4,補(bǔ)充知識(shí),3 列向量對(duì)列向量的導(dǎo)數(shù),設(shè)列向量Y中的元素都是列向量X中元素的函數(shù),第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,右兩式即是列向量Y 對(duì)列向量X的導(dǎo)數(shù),補(bǔ)充知識(shí),3 列向量對(duì)列向量的導(dǎo)數(shù),第九講 參數(shù)平差概述與原理 The Introduction and Principle

3、 of Parametric LS Adjustment,性質(zhì),1,2,3,5,4,6,復(fù)習(xí),第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,平差：在多余觀測(cè)的基礎(chǔ)上，依據(jù)一定的觀測(cè)模型，按照一定的數(shù)學(xué)原則，對(duì)觀測(cè)結(jié)果進(jìn)行合理的調(diào)整，求得一組沒(méi)有矛盾的最可靠結(jié)果，并評(píng)定精度，這種處理方法和過(guò)程稱為測(cè)量平差。,最小二乘原理,最小二乘平差計(jì)算本質(zhì)上是一個(gè)極值問(wèn)題。,觀測(cè)值的平差改正數(shù)向量,1、Introduction of Parametric Least-Squares Adjustment,2、M

4、athematics Model of Parametric Least-Squares Adjustment,3、Normal Equation,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,4、Steps of Parametric Least-Squares Adjustment,5、Some Examples,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and P

5、rinciple of Parametric LS Adjustment,C,A,B,Question：,要確定平面三角形的形狀，等精度觀 測(cè)三個(gè)內(nèi)角，求各角的最小二乘估值,這樣一個(gè)問(wèn)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？,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,L3,L1,L2,X1,X2,C,A,B,1）Perfect Condition,t unknowns, n equa

6、tions,If no errors, the equations is compatible,But errors exist, equations is incompatible.,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,2）If we have the true errors,But we cannot obtain the true errors ,the X is also

7、 not obtained,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,3）Regard X and as unknowns,append the additional condition,The additional condition is LS principle,如果我有一千萬(wàn)， 我就能買一棟房子。 我有一千萬(wàn)嗎？沒(méi)有。 所以我仍然沒(méi)有房子。 如果我有翅膀，我就能飛。 我有翅膀嗎

8、？沒(méi)有。 所以我也沒(méi)辦法飛。 如果把整個(gè)太平洋的水倒出， 也澆不熄我對(duì)你愛情的火 。 整個(gè)太平洋的水全部倒得出嗎？不能。 所以，我并不愛你。 第一次親密接觸,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,4）The Error Equations,VResidual error，殘差 Unknowns，未知參數(shù) Adjustment Values，平差值,1、Introduction of

9、 Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,4）The Error Equations,從數(shù)學(xué)角度上來(lái)看，最小二乘問(wèn)題就是求殘差二次型VTPV的極小值點(diǎn)問(wèn)題,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,4）The

10、Error Equations,更為一般的，設(shè),于是,誤差方程,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,5）Examples,5）Examples,or,5）Examples,or,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of

11、Parametric LS Adjustment,參數(shù)平差的概念,首先依據(jù)具體問(wèn)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)與全部觀測(cè)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，求出各未知參數(shù)及觀測(cè)量的最或然值，并精度估計(jì)。,1、Introduction of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,6）Note,只有存在多余觀測(cè)才會(huì)存在平差問(wèn)題。因?yàn)閮H有必需觀測(cè)時(shí)，觀測(cè)方程有唯一解。一般n表

12、示觀測(cè)值個(gè)數(shù)，t是必需觀測(cè)數(shù)，n大于t。,參數(shù)平差的未知數(shù)個(gè)數(shù)必須等于必需觀測(cè)個(gè)數(shù)。 未知參數(shù)必須函數(shù)獨(dú)立，即未知參數(shù)間不存在條件關(guān)系。 未知參數(shù)可以是間接觀測(cè)量，所以參數(shù)平差也叫間接平差 。 誤差方程可以是非線性形式。,2、Mathematics Model of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,1函數(shù)模型,對(duì)一個(gè)測(cè)量的實(shí)際問(wèn)題，如果有n個(gè)觀測(cè) 值，t個(gè)未知參數(shù)，對(duì)應(yīng)的存在n個(gè)改正數(shù),2、Mathematic

13、s Model of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,（觀測(cè)方程）,（誤差方程）,1函數(shù)模型,2、Mathematics Model of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,? Weight matrix of l (l=D-L)

14、 ? expectation of V,l=D-L,1函數(shù)模型,2、Mathematics Model of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,對(duì)于觀測(cè)值向量L的真誤差向量,1函數(shù)模型,兩邊取數(shù)學(xué)期望,2、Mathematics Model of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of

15、Parametric LS Adjustment,2隨機(jī)模型,（觀測(cè)向量的真誤差的期望為零）,（權(quán)矩陣正定）,3、Normal Equation（法方程）,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,1）The Principle of The Least Squares Estimation,可求得唯一估值,聯(lián)立,根據(jù),3、Normal Equation （法方程）,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS A

16、djustment,2）The Normal Equation,極值方程,由誤差方程得,極值方程為,誤差方程代入上式：,3、Normal Equation （法方程）,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,2）The Normal Equation,Nnormal equation coefficient matrix(系數(shù)陣） Unormal equation vector（自由項(xiàng)）,or,參數(shù)的解向量：,3、Normal Equation（法方程）,第九講 參數(shù)平差概述與原理 The

17、 Introduction and Principle of Parametric LS Adjustment,Note:,（1）法方程系數(shù)陣N的維數(shù)tt,（2）法方程的秩r(N)=r(ATPA)=r(A)=t,（3）當(dāng)r(A)=t時(shí)，稱矩陣A列滿秩，為保證A列滿秩，未知參數(shù)選 取是關(guān)鍵,（4）只有N滿秩時(shí)，法方程才有唯一解,3、Normal Equation（法方程）,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,3） is the least point of VTPV,設(shè)另有解 ，改正數(shù)向

18、量V1，只要,命題就得證,因?yàn)?3、Normal Equation,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,3） is the least point of VTPV,4） is the unbiased estimate,5） The Expectation of V is .?,4、 Steps of Parametric Least-Squares Adjustment,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,1）選取足夠且獨(dú)立的參數(shù)作為未知量(t個(gè))；,2）列出誤差方程式(n個(gè))；,3）依誤差方程式組成法方程；,4）解法方程求得未知向量估值；,5）回代X向量，求出改正數(shù)和觀測(cè)值平差值；,6）精度估計(jì)。,5、Special Example:,第九講 參數(shù)平差概述與原理 The Introduction and Principle of Parametric LS Adjustment,The Normal Equation for Independent Obs.,法方程的一般形式,觀測(cè)值獨(dú)立