平差(教学课件)-成晓倩-第3章条件平差课件

1、主讲：成晓倩主讲：成晓倩 河南理工大学测绘学院河南理工大学测绘学院平差(教学课件)-成晓倩-第3章条件平差平差(教学课件)-成晓倩-第3章条件平差数学模型数学模型条件平差条件平差函数模型函数模型随机模型随机模型0WAV0AWnnnnnnPQD,120,20,minTV PV 在数学中就是求函数的条件极值问题在数学中就是求函数的条件极值问题平差(教学课件)-成晓倩-第3章条件平差1 , nLnnP,1 ,nV1 ,nLnnLLLL211 ,nnvvvV211 ,nnnpppP21,nnLLLL211 ,nnnvLvLvLLLL221121000022110221102211rLrLrLrbLbL

2、bLbaLaLaLannnnnn平差值线性平差值线性条件方程条件方程平差(教学课件)-成晓倩-第3章条件平差000022110221102211rLrLrLrbLbLbLbaLaLaLannnnnn条件方程条件方程000221122112211rnnbnnannwvrvrvrwvbvbvbwvavava改正数条件方程改正数条件方程)()()(022110221102211rLrLrLrwbLbLbLbwaLaLaLawnnrnnbnna闭合差闭合差0WAV)(0AALW平差(教学课件)-成晓倩-第3章条件平差构造拉格构造拉格朗日函数朗日函数0WAV)(0AALW)(2WAVKPVVTTTrba

3、rkkkK1 ,乘系数乘系数022)(2)(AKPVVAVKVPVVdVdTTTT一阶导数一阶导数AKPVTTKAVPTTKAPVT1改正数方程改正数方程平差(教学课件)-成晓倩-第3章条件平差KAPVT1)(1ribiaiiikrkbkapv0WAV01WKAAPT法方程法方程TAAPN10WNKWNK1000rrbabrbaarbawkprrkpbrkparwkpbrkpbbkpabwkparkpabkpaa法方程纯量形式法方程纯量形式VLL平差(教学课件)-成晓倩-第3章条件平差KAPVT1WNK1VLL平差值平差值1()()TR NR AP Ar0WNK条件平差法方程的个数条件平差法方

4、程的个数= =多余观测个数多余观测个数r平差(教学课件)-成晓倩-第3章条件平差平差(教学课件)-成晓倩-第3章条件平差rPVVT0rPVVT20PVVT的计算：的计算：2222211.nnTvpvpvppvvPVVaKWPVVbTT.rrbbaakwkwkwKWKAVKAVKAPPVPVVTTTTTTT)()(1平差(教学课件)-成晓倩-第3章条件平差LVKWL、LL 00)(AALAALW011011)(ANALNAALNWNK0111101111)(ANAPALNAPANALNAPKAPVTTTT0111101111)()(ANAPLANAPEANAPALNAPLVLLTTTT01101

5、1010111110ANAPANAPANALANAPEANAPANAELVKWLZTTTT矩阵形式：矩阵形式：TZZLLQKQ K平差(教学课件)-成晓倩-第3章条件平差LVKWL、LLVLKLWLLLLVVVVKVWVLLKKVKKKWKLLWWVWKWWWLLLLVLKLWLLZZQQQQQQQQQQQQQQQQQQQQQQQQQQAQNQAQAQNQAQAQNQANQAQAAQNQAAQNNEAQNAQENAQAQNQAQAQNQANQAQAQTTTTTTTTTT11111111111000000LWKV、 、平差(教学课件)-成晓倩-第3章条件平差011011010111110ANAP

6、ANAPANALANAPEANAPANAELVKWLZTTTT？、VLWWQQ单独求解单独求解NAAPAAQAQAQTTLLTLLWW1)()(0)()(11111111111111111APNAAPNAPPANAPAPNAEAPNAPANAPEQANAPQTTTTTTTTVL平差(教学课件)-成晓倩-第3章条件平差),(21nLLLfFnLLnLLLLLdLfLdLfLdLfFd22111212,TnnL LL LL Lfffff ffLLLfQfQLLTFF1 ()TLLQQAQNAQ1()()TTFFQf QfAQfNAQfFFFFQD20平差(教学课件)-成晓倩-第3章条件平差0WAV

7、00 ALA)(0AALW0WNKTAAPN1WNK1KAPVT1VLL平差(教学课件)-成晓倩-第3章条件平差rPVVT02222211.nnTvpvpvppvvPVVaKWPVVbTT.),(21nLLLfFLdfLdLfLdLfLdLfFdTnLLnLLLL2211TLLnLLLLTnLfLfLfffff2121,1()()TTFFQf QfAQfNAQfFFFFQD20平差(教学课件)-成晓倩-第3章条件平差48 2436PAT157 3216L 273 0308L 3124 5128L 4104 3320L PCTPCTPCT解：解：03604321LLLL036011114321L

8、LLL平差(教学课件)-成晓倩-第3章条件平差解：解：057 32 1673 03 08()(1111360 )12124 51 28104 33 20WALA 0)21(11114321 vvvv01WKAAPT11114321,ppppPnn4( 12)0aK 3aK平差(教学课件)-成晓倩-第3章条件平差解：解： 33333111111111KAPVT11122233344457 32 3373 03 05124 51 25104 33 17LLvLLvLvLLvL1234360LLLL平差(教学课件)-成晓倩-第3章条件平差解：解：36333311113333PVVT61360 rPV

9、VT012178 5954PCPATTLL6342480011 4321 LLLLTPCTf00111()()PCTTTQf QfAQfNAQf100111111 111125. 011111111001100111111001160 PCPCTTQ平差(教学课件)-成晓倩-第3章条件平差000022110221102211rLrLrLrbLbLbLbaLaLaLannnnnn00ALAnnnnnnPQD,120,20,110r n nrAVW平差(教学课件)-成晓倩-第3章条件平差000221122112211rnnbnnannwvrvrvrwvbvbvbwvavava0WAV)(0AALW

10、KAPVT1)(1ribiaiiikrkbkapv0WNKTAAPN1平差(教学课件)-成晓倩-第3章条件平差rPVVT02222211.nnTvpvpvppvvPVVaKWPVVbTT.rrbbaakwkwkw011011010111110ANAPANAPANALANAPEANAPANAELVKWLZTTTT11111111111000000TTTTZZTTTTTTQQAQA NQA N AQQQA N AQAQNEAQQN AQENN AQQA N AQQAQA NQA N AQQQA N AQQQA N AQLWKV、 、平差(教学课件)-成晓倩-第3章条件平差),(21nLLLfFnL

11、LnLLLLLdLfLdLfLdLfFd2211fQfQLLTFF1TTFFQf QfAQfNAQfFFFFQD20ffpafpbfprfpTLLnLLLLTnLfLfLfffff2121,平差(教学课件)-成晓倩-第3章条件平差平差(教学课件)-成晓倩-第3章条件平差Condition Equations of Different Geometrical Models1. 高程控制网条件方程 Condition Equations of Height Control Network Condition Equations of Traverse Control Network Conditi

12、on Equations of Triangulation Network Condition Equations of Trilateration Network平差(教学课件)-成晓倩-第3章条件平差1.1.高程控制网条件方程高程控制网条件方程Conditions Equations of Height Control Network必要起算数据必要起算数据平差(教学课件)-成晓倩-第3章条件平差1.1.高程控制网条件方程高程控制网条件方程Conditions Equations of Height Control Network平差(教学课件)-成晓倩-第3章条件平差1.1.高程控制网条

13、件方程高程控制网条件方程Conditions Equations of Height Control Network平差(教学课件)-成晓倩-第3章条件平差2712423546757800 00 0ABhhHHhhhhhhhhhhhh271124223534674578500 00 0vvwvvvwvvvwvvvwvvvw1272124323544675578()()() ()()ABwhhHHwhhhwhhhwhhhwhhh 平差(教学课件)-成晓倩-第3章条件平差1345346712580000ACABCDhhhhHHhhhhhhHHhhHH13451346721235840000vvvv

14、wvvvvwvvwvvw1134523467312458()()()()ACABCDwhhhhHHwhhhhwhhHHwhhHH 12314253647801110000110110010000000001001vvvwvwvwvwvv平差(教学课件)-成晓倩-第3章条件平差0 00 0875764532421hhhhhhhhhhhh0 00 04875376425321421wvvvwvvvwvvvwvvv) ()( )()(8754764353224211hhhwhhhwhhhwhhhw平差(教学课件)-成晓倩-第3章条件平差ABCD1h2h3h4h5h6h1452563460 00hhh

15、hhhhhh1451256234630 00vvvwvvvwvvvw114522563346 whhhwhhhwhhh 平差值方程式平差值方程式改正数方程式改正数方程式闭闭合合差差平差(教学课件)-成晓倩-第3章条件平差已知高程值已知高程值高差观测值高差观测值mHA100.31mHB165.3411.001hm 21.002hm 11Skm22Skm31.064hm 40.500hm 32Skm41Skm50.504hm 60.060hm 52Skm62Skm70.560hm 81.000hm 72.5Skm82.5Skm平差(教学课件)-成晓倩-第3章条件平差000 083174363254

16、2BAHHhhhhhhhhhhhh00004831374326321542wvvvwvvvwvvvwvvv10000101010011000010011000011010A0422)(8317436325420BAHHhhhhhhhhhhhhAAhW平差(教学课件)-成晓倩-第3章条件平差已知高程值已知高程值高差观测值高差观测值mHA100.31mHB165.3411.001hm 21.002hm 11Skm22Skm31.064hm 40.500hm 32Skm41Skm50.504hm 60.060hm 52Skm62Skm70.560hm 81.000hm 72.5Skm82.5Skm5

17、 . 25 . 22212211P5 . 522025 . 521226201251TAAPN10.64260.10611.00100.3254KN W平差(教学课件)-成晓倩-第3章条件平差10.3254 1.07311.13900.35841.28530.21222.50260.8136TTVP A K 1.00131.0031 1.06290.50040.50270.05980.56251.0008ThhV平差(教学课件)-成晓倩-第3章条件平差平差(教学课件)-成晓倩-第3章条件平差平差(教学课件)-成晓倩-第3章条件平差平差(教学课件)-成晓倩-第3章条件平差Condition Eq

18、uations of Different Geometrical Models1. 高程控制网条件方程 Condition Equations of Height Control Network Condition Equations of Traverse Control Network Condition Equations of Triangulation Network Condition Equations of Trilateration Network平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of

19、Traverse Control NetworkT0Tn+1平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control NetworkABT0TCDT1nT()BBx y，()CCx y，T0Tn+1条件方程类型：条件方程类型：1 1个方位角符合条件；个方位角符合条件；2 2个坐标符合条件个坐标符合条件11()nnxy，11()x y，平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Contr

20、ol NetworkT0Tn+101CDnTT180) 1(180) 1(1101101nvTnTTninini0180) 1(1101CDniCDnTnvTTTi011Tnwvi)180) 1(110CDniTTnTw)(1CDnTT方位角方位角符合条件符合条件方程方程平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network01CnxxniBnxxx11iiiTSxcos iSiivSSiiijiijijiTviTvivTiTTjjj10111010180180180)c

21、os()(1iiSiiTvvSxjiiiSiiiiiSiiiijijivyvTxvTSvTTSx11cossincoscos 平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control NetworkiiSiiiiiSiiiijijivyvTxvTSvTTSx11cossincoscos ninnSinniiSiiBCniijivyyvTxvyvTxxxx1111111)(1coscos iv01Cnxx平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Co

22、ndition Equations of Traverse Control NetworkninnSinniiSiiBCniijivyyvTxvyvTxxxx1111111)(1coscos 0)(1cos1111 CnninnSixxvyyvTii0)(1cos111 xninnSiwvyyvTii)(1Cnxxxw纵坐标纵坐标附合条附合条件方程件方程平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network0)(1sin111 yninnSiwvxxvTii)(1Cnyy

23、yw横坐标横坐标附合条附合条件方程件方程0)(65.20621cos111xninnSiwvyyvTii0)(65.20621sin111yninnSiwvxxvTii平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network011Tnwvi0)(1sin111 yninnSiwvxxvTii0)(1cos111 xninnSiwvyyvTii条件方程汇总条件方程汇总平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equ

24、ations of Traverse Control Network011Tnwvi0)(1sin111 yninnSiwvxxvTii0)(1cos111 xninnSiwvyyvTii条件方程汇总条件方程汇总平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network0180)2(12nni012w

25、vni)180)2(12nwni0)(1cos111 xninnSiwvyyvTii0)(1sin111 yninnSiwvxxvTii)(1Bnxxxw)(1Bnyyyw条件方程汇总条件方程汇总内角和条内角和条件方程件方程平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network220p220SSp01p22SSprvvpvvprpvvSSS0iiSSp10平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equatio

26、ns of Traverse Control Network如图为一四等附合导线，测角中误差如图为一四等附合导线，测角中误差 , ,测边所用测距仪的标称精测边所用测距仪的标称精度公式度公式 ，已知数据和，已知数据和观测值见下表。试按条件平差法对此导观测值见下表。试按条件平差法对此导线进行平差，并评定线进行平差，并评定3 3号点的点位精度。号点的点位精度。= 2.5s= 5mm+5ppmD k m已知坐标已知坐标(m)已知方位角已知方位角B (187396.252 , 29505530.009)C (184817.605 , 29509341.482)TAB = 1614407.2TCD = 2

27、493027.9导线边长观测值导线边长观测值(m) 转折角度观测值转折角度观测值S1 = 1474.444S2 = 1424.717S3 = 1749.322S4 = 1950.4121 = 8530 21.12 = 25432 32.23 = 13104 33.34 = 27220 20.25 = 24418 30.0平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network解：未知导线点个数n 1 = 3，导线边数n = 4，观测角个数n + 1 = 5 近似计算导线边长

28、、方位角和各导线点坐标，列于表中近似坐标(m)近似方位角2 (187966.645 , 29506889.655)3 (186847.276 , 29507771.035)4 (186760.011 , 29509518.179)5 (184817.621 , 29509341.465) T1 = 67 14 28.3T2= 141 47 00.5T3 = 92 51 33.8T4= 185 11 54.0T5 = 249 30 24.0(1)组成改正数条件方程及第3点平差后坐标函数式)(51CDTTw=3.9)(42Cxxw= -1.6cm)(43Cyyw= 1.7cm改正数条件方程闭合差项

29、：平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network改正数条件方程：5114451211445131101cos()02062.651sin()02062.65iiiiiiSiSvwTvyy vwTvxx vw 123453.90vvvvv123412340.38680.78570.04990.99591.84791.18870.76140.08571.60SSSSvvvvvvvv123412340.92210.61860.99880.09061.25021.5267

30、0.98400.94171.70SSSSvvvvvvvv即：平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network0000111110.38680.78570.04990.99591.84791.18870.76140.085700.92210.61860.99880.09061.25021.52670.98400.94170A系数阵：改正数向量：123412345SSSSvvvvvVvvvv 闭合差向量：W 平差(教学课件)-成晓倩-第3章条件平差2.

31、2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network第3点坐标平差后函数式：311211122coscosxxxxxSTST311211122sinyyyyyS sinTST全微分：22331111cosiiidxTdsyyd22331111siniiidyTdsxx d30.38680.7857001.08650.4273000Txf30.92210.6186000.26620.5427000Tyf平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equat

32、ions of Traverse Control Network(2)确定边角观测值的权：设单位权中误差：02.5 根据标称精度公式计算测边中误差：655 10DkmmmD 测角观测值权：1P测边观测值权：2202cmDDP2秒311111P平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network(3)组成法方程：113.27260.59453.3563TTKN WAP AW1TVP A K0.67050.64631.02940.39830.72940.7

33、8830.65100.03533.2726T1234123451474.4507051424.7234631749.3327391950.41598385 30 22.4254 32 30.713104 34.6272 20 20.524418 36.5SSSS点坐标：点2：187966.644,29506889.663点3：186847.270,29507771.048点4：186760.000,29509518.201平差(教学课件)-成晓倩-第3章条件平差2.2.导线控制网条件方程导线控制网条件方程Condition Equations of Traverse Control Network(4)精度评定：02.6TV PVr 3111133330.6154TTxxxxxQf PfAPfNAPf单位权中误差：3034.16xxQcm 223332.46xycm 3111133330.2788TTyyyyyQf PfAPfNAPf3031.88yyQ