平差课程设计.docx
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平差课程设计
误差理论与测量平差
实习报告
班级:
滨江2013级测绘工程
组别:
第一组
组长:
杜迎,范璐(副)
组员:
任蔚,单煜,戴展鹏,欧宇杰,于纹鉴,高英杰
本题n=17,有17给误差方程,其中14个角度误差方程,3个边长误差方程,必要观测数t=2*4+6=14,现取定带定点坐标和测站定向角为参数,即
X=[X3Y3X4Y4X5Y5X6Y6Z1Z2Z3Z4Z5Z6]
一.求待定点坐标值:
1.计算近似坐标
近似坐标通过已知坐标点12,和观测方位角和方向观测值求近似坐标和近似边长。
已知坐标点
X1=121088.500;
Y1=259894.000;
X2=127990.100;
Y2=255874.600;
方向观测值
L1=72+10/60+28.4/3600=72.1746;
L2=66+27/60+28.9/3600=66.4580;
L3=123+11/60+34.1/3600=123.1928;
L4=85+13/60+37.4/3600=85.2271;
L5=79+9/60+48.7/3600=79.1635;
L6=72+24/60+56.4/3600=72.4157;
L7=88+58/60+29.5/3600=88.9749;
L8=132+23/60+35.2/3600=132.3931;
边长观测值
s1=4451.417;
s2=5564.592;
s3=5569.269;
方向角确定
f1=atand((Y2-Y1)/(X2-X1))+360=329.784;
f2=f1-L2-180=83.326;
f3=f2-L8+180=130.9329;
f4=f3-L4=45.7058;
f5=f4+180-L6=153.2901;
f6=f5+180-L5=254.1266;
f7=f6-L7-L3+180=221.9589;
近似坐标求取
x4=s2*cos(f2)+X2=128636.8171;
y4=s2*sin(f2)+Y2=261401.4835;
x6=s1*cos(f4)+x4=131745.4323;
y6=s1*sin(f4)+y4=264587.645;
x5=s3*cos(f5)+x6=126770.4392;
y5=s3*sin(f5)+x6=267090.883;
x3=125972.6859
y3=264285.4008
近似边长
s13=sqrt((y3-Y1)^2+(x3-X1)^2)=6568.0798;
s24=sqrt((y4-Y2)^2+(x4-X2)^2)=5564.5920;
s46=sqrt((y6-y4)^2+(x6-x4)^2)=4451.417;
s56=sqrt((y6-y5)^2+(x6-x5)^2)=5569.269;
s35=sqrt((y5-y3)^2+(x5-x3)^2)=2916.7003;
s43=sqrt((y4-y3)^2+(x4-x3)^2)=3926.13984;
s2-s24=0
s3-s56=0
s1-s46=0
2.计算方位角改正数系数(单位:
分米)
a=206264.8sin(f)/(s*10);
b=-206264.8cos(f)/(s*10);
a12=-1.2997a21=1.2997
b12=2.232b21=-2.232
a13=2.1a31=-2.1
b13=2.335b31=-2.335
a24=3.682a42=-3.682
b24=0.431b42=-0.431
a34=-3.969a43=3.969
b34=3.442b43=-3.442
a46=3.317a64=-3.317
b46=3.236b64=-3.236
a35=6.802a53=-6.802
b35=1.934b53=-1.934
a56=-1.665a65=1.665
b56=3.308b65=-3.308
3.确定角和边的权
设单位权中误差σ0=1.2″,角度观测值的权为Pl=σ0^2/σl^2=1;
导线边的权Ps=σ0^2/12=0.01单位:
秒平方比上毫米平方
1.常数项
边长误差方程的常数项l
方向观测值(°)
近似坐标方位角(°)
α—L
(°)
—l=α—L—Z(″)
1
2
0.0000
329.784
329.784
-0.612
3
72.1746
41.9589
329.78434
0.612
Z1
329.78417
2
4
0.0000
83.326
83.326
0.054
1
66.4580
149.7784
83.32597
-0.054
Z2
83.325985
3
1
0.0000
221.9589
221.9589
1.08
4
88.9749
310.9329
221.95804
-2.016
5
212.1677
74.1266
221.9589
1.08
Z3
221.9586
4
6
0.0000
45.7058
45.7058
0
3
85.2271
130.9329
45.7058
0
2
217.6202
263.326
45.7058
0
Z4
45.7058
0
5
6
0.0000
333.2901
254.1248
-3.24
3
79.1635
254.1266
254.1266
3.24
Z5
254.1257
6
4
0.0000
225.7058
153.2901
0
5
72.4157
153.2901
153.2901
0
Z6
153.2901
边长误差方程
边4-6V1=-0.698x4-0.7158y4+0.698x6-0.7158y6-l1
边2-4V2=0.116x4+0.993y4-l2
边6-5V3=0.893x6-0.449y6-0.893x6+0.449y6-l3
B=[00000000-100000;
-2.1-2.335000000-100000;
00-3.682-0.43100000-10000;
000000000-10000;
-2.1-2.33500000000-1000;
-3.9693.4423.969-3.442000000-1000;
6.8021.93400-6.802-1.9340000-1000;
003.3173.23600-3.3173.236000-100;
3.9693.4423.969-3.4420000000-100;
00-3.682-0.4310000000-100;
0000-1.6653.3081.6653.3080000-10;
6.8021.93400-6.802-1.934000000-10;
003.3173.23600-3.317-3.23600000-1;
0000-1.6653.3081.665-3.30800000-1;
00-0.698-0.7160.6980.71600000000;
000.1160.9930000000000;
0000-0.8730.4490.893-0.449000000;]
l=[0.612;-0.612;-0.054;0.054;-1.08;2.016;-1.08;0;0;0;3.24;-3.24;0;0;0;0;0];
p=[1;1;1;1;1;1;1;1;1;1;1;1;1;1;0.01;0.01;0.01];
P=diag(p);
NBB=B'*P*B;
W=B'*P*l;
x=inv(NBB)*W;
v=B*x-l;
结果:
由于我们系数ab用的单位是分米所以结果应该缩小100倍
所以
x3’=0.002726
y3’=0.001911
x4’=-0.009433
y4’=-0.018437
x5’=0.012872
y5’=-0.028742
x6’=0.006453
Y6’=0.048204
平差值
X3=x3+x3’=125972.6886
Y3=y3+y3’=264285.4027
X4=128636.8077
Y4=261401.4651
X5=126770.4521
Y5=267090.8543
X6=131745.4388
Y6=264587.6932
二.精度评定:
单位权方差公式
求得单位权中误差(此处用m0表示)m0=0.3401
根据矩阵,得Qx4=0.5226,Qy4=3.6605
Qx5=19.0276,Qy5=266.4110
点位中误差公式:
代入公式得:
点4的中误差m4=0.6956;点5的中误差m5=5.746。
三.误差椭圆:
根据Q矩阵可知Qx4y4=1.27,Qx5y5=-70.8276
(1)
根据位差极值公式可求得K4=4.0370,K5=285.0681
E42=0.4754,E52=32.9947,E4=0.6894,E5=5.744
F42=8.4495*10-3,F52=0.0214,F4=0.092,F5=0.146
QEE4=(QXX+QYY+K)/2=4.11005,QFF4=0.07305
根据tanΨE=(QEE-QXX)/QXY
可求得:
tanΨE4=2.8248进而求得ΨE4=79.5°或250.5°
tanΨF4=-0.354,ΨF4=-19.5°或160.5°
QEE5=(QXX+QYY+K)/2=285.25335,QFF4=0.18525
TanΨE5=-321.684,tanΨF5=0.266
E5=-89.8°或90.2°,F5=14.9°或194.9°
4、相对误差椭圆(4和5点)
X45=(X4+X5)=127703.6299
Y45=(Y4+Y5)=264246.1597
Δx45=x4-x5
Δy45=y4-y5
由协因数阵可以得到
Qx5x4=1.9217
Qy5y4=-16.6602
Qx5y4=4.6722
Qx4y5=-7.1571
QΔxΔx=Qx5x5+Qx4x4-2Qx5x4=15.7068
QΔyΔy=Qy5y5+Qy4y4-2Qy5y4=303.3919
QΔxΔy=Qx5y5+Qx5y4-Qx4y5+Qx4y4=-67.0727
由上面的公式
(1)得到
E2=108.2412E=40.4038
F=0.5338
tanΨE=(QEE-QXX)/QXY=-4.5108
进而求得ΨE4=-77.5°或102.4998°
tanΨF4=0.8376,ΨF4=39.9495°或219.9495°
4点的误差椭圆
5点的误差椭圆
4,5点相对误差椭圆
代码(4点):
functionvarargout=untitled5_OutputFcn(Xcenter,Ycenter,LongAxis,ShortAxis,Angle)
Xcenter=128636.8077;
Ycenter=261401.4651;
LongAxis=0.6894;
ShortAxis=0.092;
Angle=79.5;
t1=0:
.02:
pi;
t2=pi:
.02:
2*pi;
z1=exp(1i*t1);
z2=exp(1i*t2);
z1=(LongAxis*real(z1)+1i*ShortAxis*imag(z1))*exp(1i*(-Angle));
z2=(LongAxis*real(z2)+1i*ShortAxis*imag(z2))*exp(1i*(-Angle));
z1=z1+Xcenter+Ycenter*1i;
z2=z2+Xcenter+Ycenter*1i;
plot(z1,'r');
holdon
plot(z2,'r')
holdoff
gridon