6 附有参数的条件平差

误差方程推导

$$\mathit{A}\hat{\mathit{L}}+\mathit{B}\hat{\mathit{X}}+{\mathit{A}}_0=\mathbf{0}$$

令

$$\hat{\mathit{L}}=\mathit{L}+\mathit{V},\hat{\mathit{X}}={\mathit{X}}_0+\hat{\mathit{x}},\mathit{W}=-(\mathit{A}\mathit{L}+\mathit{B}{\mathit{X}}_0+{\mathit{A}}_0)$$

得到

$$\mathit{A}\mathit{V}+\mathit{B}\hat{\mathit{x}}-\mathit{W}=\mathbf{0}$$

仍取

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

由拉格朗日法可得基础方程

$$\mathit{V}=\mathit{Q}{\mathit{A}}^{\mathrm{T}}\mathit{k},{\mathit{B}}^{\mathrm{T}}\mathit{k}=\mathbf{0}$$

联立得块法方程

$$\left[\begin{array}{cc}\mathit{N}& \mathit{B}\\ {\mathit{B}}^{\mathrm{T}}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathit{k}\\ -\hat{\mathit{x}}\end{array}\right]=\left[\begin{array}{c}\mathit{W}\\ \mathbf{0}\end{array}\right],\mathit{N}=\mathit{A}\mathit{Q}{\mathit{A}}^{\mathrm{T}}$$

设

$${\mathit{N}}_B={\mathit{B}}^{\mathrm{T}}{\mathit{N}}^{-1}\mathit{B}$$

则

$$\{\begin{array}{l}\hat{\mathit{x}}={\mathit{N}}_B^{-1}{\mathit{B}}^{\mathrm{T}}{\mathit{N}}^{-1}\mathit{W}\\ \mathit{V}=\mathit{Q}{\mathit{A}}^{\mathrm{T}}\mathit{k}=-\mathit{Q}{\mathit{A}}^{\mathrm{T}}{\mathit{N}}^{-1}(\mathit{B}\hat{\mathit{x}}-\mathit{W})\end{array}$$

最终公式

对于闭合差 $\mathit{W}=-(\mathit{A}\mathit{L}+\mathit{B}{\mathit{X}}_0+{\mathit{A}}_0)$：

令 $\mathit{N}=\mathit{A}\mathit{Q}{\mathit{A}}^{\mathrm{T}}$，${\mathit{N}}_B={\mathit{B}}^{\mathrm{T}}{\mathit{N}}^{-1}\mathit{B}$，则有改正数向量

$$\{\begin{array}{l}\hat{\mathit{x}}={\mathit{N}}_B^{-1}{\mathit{B}}^{\mathrm{T}}{\mathit{N}}^{-1}\mathit{W}\\ \mathit{V}=\mathit{Q}{\mathit{A}}^{\mathrm{T}}\mathit{k}=-\mathit{Q}{\mathit{A}}^{\mathrm{T}}{\mathit{N}}^{-1}(\mathit{B}\hat{\mathit{x}}-\mathit{W})\end{array}$$

最终平差值

$$\{\begin{array}{l}\hat{\mathit{L}}=\mathit{L}+\mathit{V}\\ \hat{\mathit{X}}={\mathit{X}}^0+\hat{\mathit{x}}\end{array}$$

精度评定

• 单位权方差估值公式不变：${\hat{\sigma }}_0^2={\displaystyle \frac{{\mathit{V}}^{\mathrm{T}}\mathit{P}\mathit{V}}{n-t}}$，与参数改正数 $\hat{\mathit{x}}$ 无关
• 参数改正数协因数阵：${\mathit{Q}}_{\hat{x}\hat{x}}={\mathit{N}}_B^{-1}$

公式总结

• 函数模型：$\mathit{A}\mathit{V}+\mathit{B}\hat{\mathit{x}}-\mathit{W}=\mathbf{0}$
• 随机模型：$\mathit{D}={\sigma }_0^2\mathit{Q}={\sigma }_0^2{\mathit{P}}^{-1}$

平差步骤	公式
列条件方程	$\mathit{A}\mathit{V}+\mathit{B}\hat{\mathit{x}}-\mathit{W}=\mathbf{0}$
组成法方程	$\left[\begin{array}{cc}\mathit{N}& \mathit{B}\\ {\mathit{B}}^{\mathrm{T}}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathit{k}\\ -\hat{\mathit{x}}\end{array}\right]=\left[\begin{array}{c}\mathit{W}\\ \mathbf{0}\end{array}\right]$ $\mathit{N}=\mathit{A}\mathit{Q}\mathit{B}{\mathit{A}}^{\mathrm{T}}$
法方程解	$\left[\begin{array}{c}\mathit{k}\\ -\hat{\mathit{x}}\end{array}\right]={\left[\begin{array}{cc}\mathit{N}& \mathit{B}\\ {\mathit{B}}^{\mathrm{T}}& \mathbf{0}\end{array}\right]}^{-1}\left[\begin{array}{c}\mathit{W}\\ \mathbf{0}\end{array}\right]$
计算改正数	$\hat{\mathit{x}}={\mathit{N}}_B^{-1}{\mathit{B}}^{\mathrm{T}}{\mathit{N}}^{-1}\mathit{W}$ $\mathit{V}=\mathit{Q}{\mathit{A}}^{\mathrm{T}}\mathit{k}$
观测量平差值	$\hat{\mathit{X}}={\mathit{X}}^0+\hat{\mathit{x}}$ $\hat{\mathit{L}}=\mathit{L}+\mathit{V}$
单位权方差估值	${\hat{\sigma }}_0^2={\displaystyle \frac{{\mathit{V}}^{\mathrm{T}}\mathit{P}\mathit{V}}{n-t}}={\displaystyle \frac{{\mathit{V}}^{\mathrm{T}}\mathit{P}\mathit{V}}{r}}$ ${\mathit{Q}}_{\hat{x}\hat{x}}={\mathit{N}}_B^{-1}$
平差值函数的方差	${\mathit{Q}}_{\hat{F}\hat{F}}={\mathit{F}}^{\mathrm{T}}{\mathit{Q}}_{\hat{L}\hat{L}}\mathit{F}$ ${\mathit{D}}_{\hat{F}\hat{F}}={\hat{\sigma }}_0^2{\mathit{Q}}_{\hat{F}\hat{F}}$