第六章 测量误差

有极限（ 有极限（3m) 大小规律
偶然误差
符号规律 抵偿性
Lim n→∞
[∆] n
=0
5.3 误差传播定律
在未知量是通过其它观测值间接求得的情况下。 在未知量是通过其它观测值间接求得的情况下。 例如： 例如：视距测量
平距 D=kLcos2 α 高差 h= k L sin2 α + i – s 一般形式： 一般形式：Z=f ( x1, x2, x3 ············ xn ) 已知各观测值的中误差： 已知各观测值的中误差： m1, m2, m3 ············ mn 求 mz =? 取函数 Z 的全微分
2 2 2 2 2 2
2 2
2
∂F 2 ∂F 2 ∂F 2 mn m1 + m2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ mZ = ∂x ∂x ∂x 1 2 n
2
举例
例1、 设测得圆形的半径 r =1.465m，已知其中误差 m = ± 0.002m, 求 、 ， 其周长 l 及其中误差 m l 。 解： l=2πr =2π× 1.465 =9.205 又： dl=2πdr 根据误差传播定律 m l 2 = (2π)2 m 2 m l = 2π( ±0.002)=±0.013m ± 最后得 l = 9.205 ±0.013m
第6 讲 测量误差
测量误差 系统误差 偶然误差 单一观测值与真值之差 [∆∆] n 衡量一组观测值的精度
真误差： 真误差： ∆i = Li – X 中误差： m = σ`= ± 中误差： 极限误差： 极限误差： ∆极 = 3 m 容许误差： 容许误差： ∆容 = 3 m 高精度测量时 容许误差： 容许误差： ∆容 = 2 m
最或是值的计算公式为： 最或是值的计算公式为：
L0 = p 1l1 + p 2 l 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n l n [ pl ] = [p ] p1 + p 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n
L0 =
p 1l1 + p 2 l 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n l n [ pl ] = [p ] p1 + p 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n
根据误差传播定律有： 根据误差传播定律有：
1 2 1 2 1 2 n 2 M = 2 m + 2 m + ⋅⋅⋅⋅⋅⋅ 2 m = 2 m n n n n
2
m M= n
图示
M 1.8 0.8 0.6 0.4 0.2 5 10 15 20 n
m M= n
5.4 等精度直接观测平差
平差：
根据多余观测求未知量最 可靠值 最或是值）并平定精度（求中误差） （最或是值）并平定精度（求中误差）的过程。 直接平差 间接平差
k
k→∞

2
1
1
k
f2
2
[∆ x ] + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + f [∆ x ]
2
n
k
n
k

σ = f σ + f σ + ⋅⋅⋅⋅⋅⋅⋅⋅ f σ
2 2 2 2 2 2 Z 1 1 2 2 n
2 n
有限时： 当 k 有限时：
mZ = f1 m1 + f 2 m2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ f n mn
l1 + l2 + l3+ l4 4 l`1 + l`2 + l`3 3 l1 + l2 + l3+ l4 + L`1 + l`2 + l`3 7
L2=
观测值的最终结果应该是： 观测值的最终结果应该是： L= 上式实际是： 上式实际是： L=
4L1 + 3L2 4+3
4L1 + 3L2 L= 4+3 4 3 L= L1 + 7 7
2 1
(p
m1 + p 2 m 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n m n
2 2 2 2
2
) = nm
2 0
m1
2
∆ 21 = 1
2 2
mi
2
∆ 2i = 1
2 2
单一观测值的中误差
(p
2
2 1
∆1 + p2 ∆ 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ pn ∆ n
2
) = nm
)
=
[p] [p]
2
m 02 =
[p]
m 02
加权平均值的中误差
一般用观测值的改正 数计算中误差即： 数计算中误差即：
m =± =±
[v2 ] n-1
[∆2 ] =± =± n
M0
2
1 ( p1 2 m 1 2 + p 2 2 m 2 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n 2 m n 2 ) = [ p ]2
对上式两边平方求和并除以 k ：
[∆ z ] =
2
k
f
2
1
[∆ x ] + f [∆ x ] + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + f [∆ x ] +
2 2 2
1
k
2 2
2
k
2 n
n
k
i , j =1 i≠ j
∑
n
fi f j
[∆ x ∆ x ]
i j
k
[∆ z ] =
2
k
f
2
1
[∆ x ] + f [∆ x ] + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + f [∆ x ] +
m2 p1 = =4 2 m1 m2 p2 = =3 2 m2
2、权与中误差的关系 、 m 1 m = = m n 4
1
m 1 m2 = = m n 3
权与中误差成反比，中误差越大，权越小。因此，可用中误差来定义权。 权与中误差成反比，中误差越大，权越小。因此，可用中误差来定义权。
p1 =
λ
m1
2
1、等精度直接观测平差（求最或是值；评定精度） 、等精度直接观测平差（求最或是值；
真误差： 真误差： ∆1 = l1 – X ∆2 = l2 – X ············ ∆n =ln – X [∆ ] =[ l ] –[ X] n n n [∆ ] X=L- n (n→∞） ） [ X] [ l] [∆ ] n = n – n X=L 单一观测值与真值之差
1/300
例3、 对某段距离测量了 次，观测值为 1 , l2 , ··············· ln为相 、 对某段距离测量了n次 观测值为l 互独立的等精度观测值，观测中误差为m,试求其算术平均值的 互独立的等精度观测值，观测中误差为 试求其算术平均值的 中误差。 中误差。 解：
l1 + l2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ln L= n 1 1 1 dL = dl1 + dl2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ dln n n n
2
[v ] m M =± =±精度直接观测平差
1、权的概念 、
设对某量进行了两组观测 设第一组观测了4次 设第一组观测了 次 l1 , l2 , l3 , l4 设第二组观测了3次 设第二组观测了 次 l`1 , l`2 , l`3 每组分别取平均值： L1= 每组分别取平均值：
∂F ∂F ∂F dz = dx1 + dx 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + dx n ∂x 1 ∂x 2 ∂x n
用增量形式表示如下： 用增量形式表示如下：
∆ z = f 1 ∆ x1 + f 2 ∆ x 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + f n ∆ x n
设想对x 次观测： 设想对 i 进行了 k 次观测：
根据误差传播定律： 根据误差传播定律：
M0
2
1 ( p1 2 m 1 2 + p 2 2 m 2 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ p n 2 m n 2 ) = [ p ]2
因为： 因为：
pi =
M 02 =
λ
mi
[p ]
1
2
且取λ=
m02
2 2
则：
2
(p m
1
2
0
+ p2m0 + ⋅⋅⋅⋅⋅⋅ pnm0
∆i = vi + δ
对上式两边平方得： 对上式两边平方得：
i=1,2,3,---------- n
∆i ∆i = vi2 + δ2 + 2δ vi
求和
[∆2 ]= [v2 ] + nδ2 + 2δ[ v ] [∆2 ]= [v2 ] + nδ2 δ2 = (x – X) 2 =( [ l ] - X) 2 = 12 ([l ] – nX) 2 n n
用改正数计算观测值中误差 m =± ± [v2 ] n-1
求最或是值的中误差
l1 + l2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ln L= n 1 1 1 dL = dl1 + dl2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ dln n n n
根据误差传播定律有： 根据误差传播定律有：
1 2 1 2 1 2 n 2 M = 2 m + 2 m + ⋅⋅⋅⋅⋅⋅ 2 m = 2 m n n n n
2 2 2
1
2
2
2
n
k
2
k
n
k
i , j =1 i≠ j