10 附有限制条件的间接平差

模型

在间接平差基础上，额外获知一些参数之间的关系时，即变为附有限制条件的间接平差。

• 观测值个数：$n$
• 参数个数：$u$
• 参数条件数：$s$
• 必要观测数：$t=u-s$
• 多余观测数：$r=n-t=n-u+s$

$$\{\begin{array}{l}\hat{\mathit{L}}=\mathit{B}\hat{\mathit{X}}+\mathit{d}\\ \mathit{C}\hat{\mathit{X}}+{\mathit{C}}_0=\mathbf{0}\end{array}$$

令

$$\hat{\mathit{X}}={\mathit{X}}_0+\hat{\mathit{x}},\mathit{l}=\mathit{L}-\mathit{B}{\mathit{X}}_0-\mathit{d},{\mathit{W}}_x=-(\mathit{C}{\mathit{X}}_0+{\mathit{C}}_0)$$

得

$$\{\begin{array}{l}\mathit{V}=\mathit{B}\hat{\mathit{x}}-\mathit{l}\\ \mathit{C}\hat{\mathit{x}}-{\mathit{W}}_x=\mathbf{0}\end{array}$$

解算

目标函数取最小：

$${\mathit{V}}^{\mathrm{T}}\mathit{P}\mathit{V}=min$$

在约束 $\mathit{V}=\mathit{B}\hat{\mathit{x}}-\mathit{l}$ 下，设联系数向量 ${\mathit{k}}_{n\times 1}$ 与 ${\mathit{g}}_{n\times 1}$，得拉格朗日条件极值函数：

$$\mathit{\Phi }={\mathit{V}}^{\mathrm{T}}\mathit{P}\mathit{V}-2{\mathit{k}}^{\mathrm{T}}(\mathit{C}\hat{\mathit{x}}-{\mathit{W}}_x)-2{\mathit{g}}^{\mathrm{T}}(\mathit{B}\hat{\mathit{x}}-\mathit{V}-\mathit{l})=min$$$$\begin{array}{rl}\frac{\mathrm{d}\mathit{\Phi }}{\mathrm{d}\mathit{V}}& =2{\mathit{V}}^{\mathrm{T}}\mathit{P}-2{\mathit{g}}^{\mathrm{T}}=\mathbf{0}\Rightarrow \mathit{g}=\mathit{P}\mathit{V}\\ \frac{\mathrm{d}\mathit{\Phi }}{\mathrm{d}\hat{\mathit{x}}}& =2{\mathit{k}}^{\mathrm{T}}\mathit{C}+2{\mathit{g}}^{\mathrm{T}}\mathit{B}=\mathbf{0}\Rightarrow {\mathit{B}}^{\mathrm{T}}\mathit{g}+{\mathit{C}}^{\mathrm{T}}\mathit{k}=\mathbf{0}\end{array}\Rightarrow {\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{V}+{\mathit{C}}^{\mathrm{T}}\mathit{k}=\mathbf{0}$$$$\{\begin{array}{l}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{V}+{\mathit{C}}^{\mathrm{T}}\mathit{k}=\mathbf{0}\\ \mathit{V}=\mathit{B}\hat{\mathit{x}}-\mathit{l}\\ \mathit{C}\hat{\mathit{x}}-{\mathit{W}}_x=\mathbf{0}\end{array}\Rightarrow \{\begin{array}{l}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{B}\hat{\mathit{x}}+{\mathit{C}}^{\mathrm{T}}\mathit{k}={\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{l}\\ \mathit{C}\hat{\mathit{x}}-{\mathit{W}}_x=\mathbf{0}\end{array}$$

整理得到块法方程

$$\left[\begin{array}{cc}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{B}& {\mathit{C}}^{\mathrm{T}}\\ \mathit{C}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\hat{\mathit{x}}\\ \mathit{k}\end{array}\right]=\left[\begin{array}{c}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{l}\\ {\mathit{W}}_x\end{array}\right]$$

解出 $\hat{\mathit{x}}$ 后仍有

$$\mathit{V}=\mathit{B}\hat{\mathit{x}}-\mathit{l},\hat{\mathit{X}}={\mathit{X}}_0+\hat{\mathit{x}},\hat{\mathit{L}}=\mathit{L}+\mathit{V}$$

附有限制条件的间接平差精度评定推导涉及分块矩阵求逆，最终公式复杂，此处不再展开。