測邊網(wǎng)測角網(wǎng)導(dǎo)線網(wǎng)典型計(jì)算

1、目錄摘 要1第1章 測邊網(wǎng)坐標(biāo)平差21.1 近似坐標(biāo)計(jì)算31.2 計(jì)算誤差方程的系數(shù)及常數(shù)項(xiàng)41.3 誤差方程41.4 計(jì)算觀測值的權(quán)51.5 組成法方程51.6 平差值計(jì)算5 坐標(biāo)平差值5 邊長平差值計(jì)算61.7 精度計(jì)算6 單位權(quán)中誤差6 待定點(diǎn)坐標(biāo)中誤差6第2章 三角網(wǎng)坐標(biāo)平差72.1 測角網(wǎng)函數(shù)模型82.2 坐標(biāo)方位角計(jì)算92.2.1 近似坐標(biāo)方位角計(jì)算92.2.2 坐標(biāo)方位角計(jì)算92.3 近似坐標(biāo)增量、近似邊長與誤差方程系數(shù)92.4 誤差方程的組成102.5 確定權(quán)和組成法方程112.6 法方程系數(shù)陣的逆陣與參數(shù)改正數(shù)112.7 平差值計(jì)算及精度評定112.7.1 待定點(diǎn)的最或然值1

2、12.7.2 觀測值的改正數(shù)122.7.3 點(diǎn)位中誤差122.7.4 觀測值平差值12第3 章 導(dǎo)線網(wǎng)間接平差133.1 計(jì)算各邊坐標(biāo)方位角改正數(shù)方程的系數(shù)143.2 確定角和邊的權(quán)153.3 計(jì)算角度和邊長誤差方程系數(shù)和常數(shù)項(xiàng)163.4 誤差方程的組成和解163.5 平差值計(jì)算213.6 精度計(jì)算213.6.1 單位權(quán)中誤差213.6.2 待定點(diǎn)點(diǎn)位中誤差計(jì)算21參考文獻(xiàn)22指導(dǎo)老師評語23摘 要本課程設(shè)計(jì)介紹了綜合運(yùn)用測量平差基礎(chǔ)知識(shí)來解決測邊網(wǎng)、三角網(wǎng)、導(dǎo)線網(wǎng)坐標(biāo)平差及精度評定問題，先設(shè)定未知參數(shù)，根據(jù)空間幾何關(guān)系找出相應(yīng)的平差模型，按照間接平差的原理，列出觀測值誤差方程，求出法方程的各

3、系數(shù)矩陣，解算未知參數(shù)、觀測值的改正數(shù)，最后進(jìn)行精度評定，完成課題要求。通過這一課題，拓寬我們測量數(shù)據(jù)處理的知識(shí)面，啟發(fā)我們處理實(shí)際生產(chǎn)問題的新思路，針對某一實(shí)際問題，用經(jīng)典的誤差理論和比較前沿的數(shù)據(jù)處理方法進(jìn)行合理的求解，以鞏固和加強(qiáng)我們對誤差理論和現(xiàn)代測量數(shù)據(jù)處理方法的理解，增強(qiáng)我們用所學(xué)的理論方法解決實(shí)際問題的能力。關(guān)鍵字：平差 誤差方程 法方程 平差值 精度評定第1章 測邊網(wǎng)坐標(biāo)平差1.有測邊網(wǎng)如下圖所示。網(wǎng)中A、B、C及D為已知點(diǎn)，P1、P2、P3及P4為待定點(diǎn)，現(xiàn)用某測距儀觀測了13條邊長，測距精度起算數(shù)據(jù)及觀測邊長見表7-17。試按間接平差法求待定點(diǎn)坐標(biāo)平差值及其中誤差。起算數(shù)據(jù)

4、和觀測數(shù)據(jù)點(diǎn)名坐標(biāo)（m）邊長（m）坐標(biāo)方位角º "XYA53743.13661003.8267804.558138 00 08.6B47943.00266225.854C40049.22953782.7907889.381113 19 50.8D36924.72861027.086編號(hào)邊觀測值（m）編號(hào)邊觀測值（m）編號(hào)邊觀測值（m）15760.70568720.162115487.07325187.34275598.570128884.58937838.88087494.881137228.36745483.15897493.32355731.788105438.380解

5、：由題意有n=13,t=8選待定點(diǎn)的坐標(biāo)為參數(shù)，即 1.1 近似坐標(biāo)計(jì)算的近似坐標(biāo)由已知點(diǎn)A、B和觀測邊交會(huì)計(jì)算得。如下圖中設(shè)h為三角形底邊AB上的高，S為S2在A上的投影。得 m m所以待定點(diǎn)的近似坐標(biāo)為同理由及S3、S4交會(huì)計(jì)算點(diǎn)的近似坐標(biāo)；由P1、P2及S10、S5交會(huì)計(jì)算P3點(diǎn)的近似坐標(biāo)；由P1、P3及S12、S9交會(huì)計(jì)算P4點(diǎn)的近似坐標(biāo)。其近似坐標(biāo)結(jié)果為： 1.2 計(jì)算誤差方程的系數(shù)及常數(shù)項(xiàng)按公式：計(jì)算誤差方程的系數(shù)和常數(shù)項(xiàng)，由已知點(diǎn)坐標(biāo)和待定點(diǎn)近似坐標(biāo)計(jì)算系數(shù)及常數(shù)項(xiàng)結(jié)果見表1-1：表1-1 誤差方程系數(shù)及常數(shù)項(xiàng)邊號(hào)方向（m）（m）（m）a=-/b=-/1P1B-637.26857

6、25.3495760.7058310.1106-0.9939-0.00832P1A5162.866503.3215187.342033-0.9953-0.0970-0.00033P2A5061.7465985.5467838.879607-0.6457-0.7636-0.00394P2P1-101.1205482.2255483.1575030.0184-0.9998-0.00505P2P3-4914.1662950.3165731.7878520.8574-0.51470.00156P2C-8632.161-1235.4908720.1283860.98990.14170.33617P3C-

7、3717.995-4185.8065598.6122110.66410.7477-0.42218P3D-6842.4963058.4907494.9391320.9129-0.4081-0.58139P3P4-2924.0066899.2847493.3257500.3902-0.9207-0.027510P3P14813.0462531.9095438.379812-0.8850-0.46560.001911P4D-3918.490-3840.7915486.9152890.71420.70001.577112P4P17737.052-4367.3728884.588445-0.87080.

8、49160.005613P4B7099.7841357.9777228.487697-0.9822-0.1879-1.20701.3 誤差方程根據(jù)表中的a、b系數(shù)及常數(shù)項(xiàng)可列出全網(wǎng)的誤差方程，其系數(shù)和常數(shù)項(xiàng)結(jié)果列于表1-2中，共13個(gè)誤差方程：表1-2 系數(shù)、常數(shù)項(xiàng)、改正數(shù)及邊長平差值列表編號(hào)(a)/(b)/(c)/(d)/(e)/(f)/(g)/(h)/-(dm)v(dm)（m）10.1106-0.99390.00000.00000.00000.00000.00000.00000.00830.05365760.7112-0.9953-0.09700.00000.00000.00000.000

9、00.00000.00000.00030.02155187.34430.00000.0000-0.6457-0.76360.00000.00000.00000.00000.0039-0.0207838.8784-0.01840.99980.0184-0.99980.00000.00000.00000.00000.0050-0.14405483.14350.00000.00000.8574-0.5147-0.85740.51470.00000.0000-0.00150.25055731.81360.00000.00000.98990.14170.00000.00000.00000.0000-0.

10、3361-0.40308720.08870.00000.00000.00000.0000066410.74770.00000.00000.42210.29505598.64280.00000.00000.00000.00000.9129-0.40810.00000.00000.58130.20337494.95990.00000.00000.00000.00000.3902-0.9207-0.39020.92070.02750.30617493.356100.88500.46560.00000.0000-0.8850-0.46560.00000.0000-0.00190.20585438.40

11、0110.00000.00000.00000.00000.00000.00000.71420.7000-1.5771-0.12505486.903120.8708-0.49160.00000.00000.00000.0000-0.87080.4916-0.0056-0.38508884.550130.00000.00000.00000.00000.00000.0000-0.9822-0.18791.20700.00637228.4881.4 計(jì)算觀測值的權(quán)將上表中的邊長觀測值代入測距精度公式: ，算的各邊的測距精度，并設(shè)，由此算得各條邊的權(quán)，其結(jié)果均列于下表中。表1-3 各邊測距精度及權(quán)邊號(hào)1

12、2345678910111213（mm）8.88.210.88.58.711.78.610.510.58.48.511.910.21.291.490.861.381.320.731.350.910.911.421.380.710.961.5 組成法方程，及其解、列于下表中，表1-4 坐標(biāo)平差值a/b/c/d/e/f/g/h/-0.0167-0.0474-0.08160.1004-0.35080.14161.02581.0276a3.14280.2578-0.00050.0254-1.1122-0.5851-0.53840.30390.0052b0.25783.14720.0254-1.3794

13、-0.5851-0.30780.3039-0.17160.0031c-0.00050.02542.0447-0.0815-0.97040.58250.00000.00000.2466d0.0254-1.3794-0.08152.24520.5825-0.34970.00000.00000.0432e-1.1122-0.5851-0.97040.58253.57490.0070-0.13860.3269-0.8750f-0.5851-0.30780.5825-0.34970.00702.33520.3269-0.7714-0.1870g-0.53840.30390.00000.0000-0.13

14、860.32692.30700.23622.6988h0.3039-0.17160.00000.00000.3269-0.77140.23621.65311.7201求，列于表中。表1-5 參數(shù)的協(xié)因數(shù)=0.4210-0.05890.0794-0.08280.17380.00200.1300-0.1355-0.05890.5146-0.06450.3540-0.03140.2010-0.13060.18290.0794-0.06450.6706-0.11750.2380-0.25430.0980-0.2011-0.08280.3540-0.11750.73745-0.14570.25752-0

15、.13370.220100.17382-0.03140.2380-0.14570.4393-0.12260.1084-0.19480.00200.2010-0.25430.2575-0.1230.7157-0.17620.40390.1300-0.13060.0980-0.13370.1084-0.17620.53477-0.2175-0.13550.1829-0.20110.2201-0.19480.4039-0.21750.90691.6 平差值計(jì)算 1.6.1坐標(biāo)平差值 按公式計(jì)算:=48580.268m =60500.500m=48681.382m =55018.290m=43767.

16、189m =57968.610m=40843.321m =64867.980m1.6.2邊長平差值計(jì)算 按公式計(jì)算:=5760.711m =5187.344m=7838.878m =5483.143m=5731.813m =8719.088m=5598.642m =7494.959m=7493.356m =5438.400m=5486.903m =8884.550m=7228.488m1.7 精度計(jì)算1.7.1單位權(quán)中誤差：=0.36dm1.7.2待定點(diǎn)坐標(biāo)中誤差：由參數(shù)的協(xié)因數(shù)陣（即）取得參數(shù)的權(quán)倒數(shù)，計(jì)算待定點(diǎn)坐標(biāo)點(diǎn)點(diǎn)位中誤差:dm dmdmdm dmdmdm dmdmdm dmdm第2章

17、 三角網(wǎng)坐標(biāo)平差2.在下圖所示的測角網(wǎng)中，A,B,C為已知點(diǎn)，,為待定點(diǎn)，S1S10為角度觀測值，已知點(diǎn)坐標(biāo)與待定點(diǎn)近似坐標(biāo)為：點(diǎn)號(hào)已知坐標(biāo)/m點(diǎn)號(hào)近似坐標(biāo)/mXYXYA883.2892259.1385P1777.416320.647B640.2838144.1899P2844.971504.160C612.0508463.8277同精度觀測值為：編號(hào)觀測值º "編號(hào)觀測值º "155 28 13.2659 57 57.2297 41 53.9769 19 22.1393 02 06.0899 56 38.2444 03 51.6929 05 51.35

18、50 42 44.31050 57 29.0是按坐標(biāo)平差法求：（1） 誤差方程及法方程；（2） 待定點(diǎn)最或是坐標(biāo)及點(diǎn)位中誤差；（3） 觀測值改正值及平差值。解：由題意有：t=4,設(shè)待定點(diǎn),的坐標(biāo)的平差值為參數(shù)，即，。2.1 測角網(wǎng)函數(shù)模型： 其中，,將以上式子帶入并線性化得：2.2 坐標(biāo)方位角計(jì)算 2.2.1 近似坐標(biāo)方位角計(jì)算按公式得：近似坐標(biāo)方位角方向近似坐標(biāo)方位角º "方向近似坐標(biāo)方位角º "329 50 41.869 47 25.0232 08 51.8278 53 17.9139 06 45.0189 49 26.0 2.2.2 坐標(biāo)方位角計(jì)算

19、按公式得：，2.3 近似坐標(biāo)增量、近似邊長與誤差方程系數(shù)按公式， ,，得：誤差方程系數(shù)表方向-61.5085105.873214992.43005-8.4623-14.5660-176.4571-137.132249942.34842-7.28785.6636143.1807-165.365247846.362226.17257.1289183.513067.555038240.699199.8984-3.6438-245.021538.318261503.81991-8.2173-1.2851-40.3323-232.920255878.51399-1.48888.59782.4 誤差方程的

20、組成按公式 得：誤差方程系數(shù)表和常數(shù)項(xiàng) 參數(shù)角號(hào)abcd18.462314.56600.00000.00000.82-1.1745-20.22960.00000.0000-3.93-13.4603-1.46530.00000.00000.846.17257.12890.00000.00001.65-6.1725-7.1289-1.48888.5978-3.369.8984-3.6438-8.4096-4.95401.87-3.725910.77279.8984-3.6438-2.1818.360710.9222-9.89843.64385.09-9.89843.64381.6811-4.928

21、91.610-8.4623-14.56608.21731.2851-5.1因此誤差方程為：2.5 確定權(quán)和組成法方程因?yàn)槭堑染扔^測即：令 ， 由得法方程：-=02.6 法方程系數(shù)陣的逆陣與參數(shù)改正數(shù)=cm2.7 平差值計(jì)算及精度評定2.7.1 待定點(diǎn)的最或然值為：即，。 2.7.2 觀測值的改正數(shù)由得：-=（秒） 2.7.3 點(diǎn)位中誤差：， 2.7.4 觀測值平差值由公式得觀測值平差值： 改正數(shù)(秒)第3 章 導(dǎo)線網(wǎng)間接平差3、有導(dǎo)線網(wǎng)如圖所示，A、B、C、D為已知點(diǎn)，P1P6為待定點(diǎn)，觀測了14個(gè)角度和9條邊長。已知測角中誤差=10"，測邊中誤差Si=（mm）(i=1,2,9),

22、Si以m為單位，已知點(diǎn)數(shù)據(jù)和待定點(diǎn)近似坐標(biāo)為：點(diǎn)號(hào)坐標(biāo)點(diǎn)號(hào)坐標(biāo)X/mY/mPX/mY/mA871.1893220.82231825.810272.250B632.2173179.48112740.107312.579C840.9400533.40183768.340392.230D663.4752570.71004732.041470.8855681.630279.3006674.567506.177觀測數(shù)據(jù)為：編號(hào)角度觀測值º "編號(hào)角度觀測值º "編號(hào)邊長觀測值/m1301 36 31.08145 58 18.1168.5822203 22 35.2

23、9125 09 37.5294.740395 41 09.110118 35 26.3384.5234224 17 27.411131 18 15.2486.668595 05 02.11268 22 31.65125.6516318 16 06.513146 19 37.16111.449753 51 08.714139 09 15.9767.289867.456965.484設(shè)待定點(diǎn)坐標(biāo)為參數(shù)，試按間接平差法求：（1） 誤差方程；（2） 待定點(diǎn)坐標(biāo)平差值及點(diǎn)位中誤差。解：n=23,有23個(gè)誤差方程，其中14個(gè)角度誤差方程，9個(gè)邊長誤差方程，必要觀測數(shù)t=2×6=12，r=n-t=

24、11,現(xiàn)選取待定點(diǎn)坐標(biāo)平差值為參數(shù)，即：3.1 計(jì)算各邊坐標(biāo)方位角改正數(shù)方程的系數(shù) 見表3-1：表3-1坐標(biāo)方位角改正數(shù)方程系數(shù)計(jì)算表邊號(hào)方向近似邊長近似坐標(biāo)方位角° "(mm)a ("/mm)b("/mm)S1pA-51.427745.37934704.08919668.58636311 25 29.52-0.66160.7498-4.36-2.2550-1.9898S240.329-85.7038971.4324594.71765154 47 59.550.9048-0.425822.350.92721.9704S379.65128.2337141.

25、3840984.5067170 28 57.64-0.3341-0.942516.292.3006-0.8155S478.655-36.2997504.226486.62694114 46 23.350.4190-0.908041.062.16200.9977S562.5168108.89915767.34248125.5680829 51 33.47-0.8673-0.497982.920.8178-1.4246S6-99.8189-49.412712405.42772111.37966243 39 48.630.443600.896269.34-1.65970.8216S7-33.279-

26、58.4774527.0513767.28337209 38 38.440.86910.49465.63-1.51632.6644S835.292-57.4744548.7859467.44469148 26 52.550.8522-0.523311.311.60032.6062S964.533-11.09184287.53611665.4792899 45 09.320.1694-0.98554.723.10460.53363.2 確定角和邊的權(quán)因?yàn)闇y角中誤差，測邊中誤差（mm）(i=1,2,9), 以m為單位，設(shè)單位權(quán)中誤差為，則角度觀測值的權(quán)均為： 各邊的權(quán)計(jì)算式為: 。所以各邊的權(quán)為,

27、 =1.4581、=1.0555、=1.1831、=1.1538、=0.7959、 =0.8973、 =1.4861、=1.4824、=1.5271。即權(quán)陣為： 3.3 計(jì)算角度和邊長誤差方程系數(shù)和常數(shù)項(xiàng)，結(jié)果見 表3-2：表32 誤差方程系數(shù)、常數(shù)項(xiàng)角角1-2.2550-1.9898-5.2423.18223.9602-0.9272-1.97045.173-0.9272-1.97043.22781.1550-2.30060.815511.014-2.30060.81554.46250.1823-2.1620-0.99771.695-2.1620-0.99772.9798-0.4269-8.026-0.81781.42460.367-1.65970.821613.358-1.51632.66443.1760-3.4860-31.7190.92721.97040.5891-4.6348-1.51632.664416.39100.78254.0308-1.6003-2.60627.2211-1.6003-2.60624.70493.1398-1.5712-3.1046-0.53361.31132.16200.9977-3.7623-3.60391.60032.60626.314-