9 间接平差的应用

测边网间接平差

以边长为观测值、以未知点纵横坐标为参数，可列立方程。

单测边误差方程

单未知点

参数：坐标 $X_{k}, Y_{k}$
近似值：坐标 $X_{k}^{0}, Y_{k}^{0}$ ，边长 $L_{j k}^{0}$

有观测方程

{\hat{L}}_{j k} = \sqrt{({\hat{X}}_{k} - X_{j})^{2} + ({\hat{Y}}_{k} - Y_{j})^{2}}

记 ${\hat{X}}_{k} = X_{k}^{0} + {\hat{x}}_{k}$ ， ${\hat{Y}}_{k} = Y_{k}^{0} + {\hat{y}}_{k}$ ，有线性化

\begin{aligned} {\hat{L}}_{j k} & = L_{j k}^{0} + \frac{\partial {\hat{L}}_{j k}}{\partial {\hat{X}}_{k}} {\hat{x}}_{k} + \frac{\partial {\hat{L}}_{j k}}{\partial {\hat{Y}}_{k}} {\hat{y}}_{k} \\ = L_{j k}^{0} + \frac{X_{k}^{0} - X_{j}}{L_{j k}^{0}} {\hat{x}}_{k} + \frac{Y_{k}^{0} - Y_{j}}{L_{j k}^{0}} {\hat{y}}_{k} \\ = L_{j k}^{0} + {\hat{x}}_{k} \cos α_{j k}^{0} + {\hat{y}}_{k} \sin α_{j k}^{0} \end{aligned}

双未知点

参数：坐标 $X_{j}, Y_{j}, X_{k}, Y_{k}$
近似值：坐标 $X_{j}^{0}, Y_{j}^{0}, X_{k}^{0}, Y_{k}^{0}$ ，边长 $L_{j k}^{0}$

有观测方程

{\hat{L}}_{j k} = \sqrt{({\hat{X}}_{k} - {\hat{X}}_{j})^{2} + ({\hat{Y}}_{k} - {\hat{Y}}_{j})^{2}}

记 ${\hat{X}}_{j} = X_{j}^{0} + {\hat{x}}_{j}$ ， ${\hat{Y}}_{j} = Y_{j}^{0} + {\hat{y}}_{j}$ ， ${\hat{X}}_{k} = X_{k}^{0} + {\hat{x}}_{k}$ ， ${\hat{Y}}_{k} = Y_{k}^{0} + {\hat{y}}_{k}$ ，有线性化

\begin{aligned} {\hat{L}}_{j k} & = L_{j k}^{0} + \frac{\partial {\hat{L}}_{j k}}{\partial X_{j}} {\hat{x}}_{j} + \frac{\partial {\hat{L}}_{j k}}{\partial {\hat{Y}}_{j}} {\hat{y}}_{j} + \frac{\partial {\hat{L}}_{j k}}{\partial {\hat{X}}_{k}} {\hat{x}}_{k} + \frac{\partial {\hat{L}}_{j k}}{\partial {\hat{Y}}_{k}} {\hat{y}}_{k} \\ = L_{j k}^{0} + \frac{X_{k}^{0} - X_{j}^{0}}{L_{j k}^{0}} ({\hat{x}}_{k} - {\hat{x}}_{j}) + \frac{Y_{k}^{0} - Y_{j}^{0}}{L_{j k}^{0}} ({\hat{y}}_{k} - {\hat{y}}_{j}) \\ = L_{j k}^{0} + ({\hat{x}}_{k} - {\hat{x}}_{j}) \cos α_{j k}^{0} + ({\hat{x}}_{k} - {\hat{x}}_{j}) \sin α_{j k}^{0} \end{aligned}

测边网求解

\begin{aligned} {\hat{L}}_{1} & = \sqrt{(X_{A} - {\hat{X}}_{2})^{2} + (Y_{A} - {\hat{Y}}_{2})^{2}} \\ {\hat{L}}_{2} & = \sqrt{(X_{B} - {\hat{X}}_{2})^{2} + (Y_{B} - {\hat{Y}}_{2})^{2}} \\ {\hat{L}}_{3} & = \sqrt{({\hat{X}}_{1} - {\hat{X}}_{2})^{2} + ({\hat{Y}}_{1} - {\hat{Y}}_{2})^{2}} \\ {\hat{L}}_{4} & = \sqrt{(X_{A} - {\hat{X}}_{1})^{2} + (Y_{A} - {\hat{Y}}_{1})^{2}} \\ {\hat{L}}_{5} & = \sqrt{(X_{B} - {\hat{X}}_{1})^{2} + (Y_{B} - {\hat{Y}}_{1})^{2}} \end{aligned}

线性化得到 $\hat{L} = B \hat{X} + d$

[\begin{matrix} {\hat{L}}_{1} \\ {\hat{L}}_{2} \\ {\hat{L}}_{3} \\ {\hat{L}}_{4} \\ {\hat{L}}_{5} \end{matrix}] = [\begin{matrix} 0 & 0 & \cos α_{A 2}^{0} & \sin α_{A 2}^{0} \\ 0 & 0 & \cos α_{B 2}^{0} & \sin α_{B 2}^{0} \\ \cos α_{21}^{0} & \sin α_{21}^{0} & - \cos α_{21}^{0} & - \sin α_{21}^{0} \\ \cos α_{A 1}^{0} & \sin α_{A 1}^{0} & 0 & 0 \\ \cos α_{B 1}^{0} & \sin α_{B 1}^{0} & 0 & 0 \end{matrix}] [\begin{matrix} {\hat{x}}_{1} \\ {\hat{y}}_{1} \\ {\hat{x}}_{2} \\ {\hat{y}}_{2} \end{matrix}] + [\begin{matrix} L_{A 2}^{0} \\ L_{B 2}^{0} \\ L_{21}^{0} \\ L_{A 1}^{0} \\ L_{B 1}^{0} \end{matrix}]

边长误差方程的特点：

两待定点的坐标参数的系数等值反号
已知点坐标系数不出现在设计矩阵中
$j, k$ 均为已知点时，无误差方程
$j, k$ 边的误差方程， $j \to k$ 或 $k \to j$ 列立，其结果相同

精度评定

{\hat{σ}}_{0}^{2} = \frac{V^{T} P V}{n - t} = \frac{V^{T} P V}{r} Q_{\hat{x} \hat{x}} = N^{- 1}

函数式
$\hat{F} = {\hat{L}}_{j k} = \sqrt{({\hat{X}}_{k} - {\hat{X}}_{j})^{2} + ({\hat{Y}}_{k} - {\hat{Y}}_{j})^{2}}$
权函数式
$\begin{matrix} d \hat{F} = - \frac{Δ X_{j k}^{0}}{L_{j k}^{0}} d {\hat{x}}_{j} - \frac{Δ Y_{j k}^{0}}{L_{j k}^{0}} d {\hat{y}}_{j} + \frac{Δ X_{j k}^{0}}{L_{j k}^{0}} d {\hat{x}}_{k} + \frac{Δ Y_{j k}^{0}}{L_{j k}^{0}} d {\hat{y}}_{k} = f^{T} d \hat{x} \\ Q_{\hat{F} \hat{F}} = f^{T} Q_{\hat{x} \hat{x}} f \\ {\hat{σ}}_{\hat{F}} = {\hat{σ}}_{0} \sqrt{Q_{\hat{F} \hat{F}}} \end{matrix}$
相对精度
$\frac{{\hat{σ}}_{\hat{F}}}{\hat{F}} = \frac{{\hat{σ}}_{0} \sqrt{Q_{\hat{F} \hat{F}}}}{{\hat{L}}_{j k}}$

方向观测网

陀螺经纬仪可以直接观测方位角。

设未知点的坐标 $C (X_{1}, Y_{1})$ ， $D (X_{2}, Y_{2})$ 为参数，对于点 $j$ 与点 $k$ 组成的边 $j k$ 有观测方程

{\hat{α}}_{j k} = \arctan \frac{{\hat{Y}}_{k} - {\hat{Y}}_{j}}{{\hat{X}}_{k} - {\hat{X}}_{j}}

进行线性化

\begin{aligned} α_{j k}^{0} & = \arctan \frac{Y_{k}^{0} - Y_{j}^{0}}{X_{k}^{0} - X_{j}^{0}} \\ a_{j k} & = {(\frac{\partial {\hat{α}}_{j k}}{\partial {\hat{X}}_{j}})}_{0} = \frac{\sin α_{j k}^{0}}{L_{j k}^{0}} ρ^{″} \\ b_{j k} & = {(\frac{\partial {\hat{α}}_{j k}}{\partial {\hat{X}}_{j}})}_{0} = - \frac{\cos α_{j k}^{0}}{L_{j k}^{0}} ρ^{″} \end{aligned}

得到

{\hat{α}}_{j k} = α_{j k}^{0} + a_{j k} {\hat{x}}_{j} + b_{j k} {\hat{y}}_{j} - a_{j k} {\hat{y}}_{k} - b_{j k} {\hat{y}}_{k}

测角网

设未知点的坐标 $C (X_{1}, Y_{1})$ ， $D (X_{2}, Y_{2})$ 为参数。有观测方程

\begin{aligned} {\hat{L}}_{1} & = {\hat{α}}_{A B} - {\hat{α}}_{A C} = α_{A B} - \arctan \frac{{\hat{Y}}_{1} - Y_{A}}{{\hat{X}}_{1} - X_{A}} \\ {\hat{L}}_{2} & = {\hat{α}}_{A C} - {\hat{α}}_{A D} = \arctan \frac{{\hat{Y}}_{1} - Y_{A}}{{\hat{X}}_{1} - X_{A}} - \arctan \frac{{\hat{Y}}_{2} - Y_{A}}{{\hat{X}}_{2} - X_{A}} \end{aligned}

单角线性化

对于 $L_{i} = ∠ h j k$ 的线性化：

{\hat{L}}_{i} = {\hat{α}}_{j k} - {\hat{α}}_{j h}

\begin{aligned} α_{j k}^{0} & = \arctan \frac{Y_{k}^{0} - Y_{j}^{0}}{X_{k}^{0} - X_{j}^{0}} \\ a_{j k} & = {(\frac{\partial {\hat{α}}_{j k}}{\partial {\hat{X}}_{j}})}_{0} = \frac{\sin α_{j k}^{0}}{L_{j k}^{0}} ρ^{″} \\ b_{j k} & = {(\frac{\partial {\hat{α}}_{j k}}{\partial {\hat{X}}_{j}})}_{0} = - \frac{\cos α_{j k}^{0}}{L_{j k}^{0}} ρ^{″} \end{aligned}

\begin{aligned} {\hat{α}}_{j k} = \arctan \frac{{\hat{Y}}_{k} - {\hat{Y}}_{j}}{{\hat{X}}_{k} - {\hat{X}}_{j}} & = α_{j k}^{0} + a_{j k} {\hat{x}}_{j} + b_{j k} {\hat{y}}_{j} - a_{j k} {\hat{x}}_{k} - b_{j k} {\hat{y}}_{k} \\ {\hat{α}}_{j h} = \arctan \frac{{\hat{Y}}_{h} - {\hat{Y}}_{j}}{{\hat{X}}_{h} - {\hat{X}}_{j}} & = α_{j h}^{0} + a_{j h} {\hat{x}}_{j} + b_{j h} {\hat{y}}_{j} - a_{j h} {\hat{x}}_{h} - b_{j h} {\hat{y}}_{h} \end{aligned}

代入 ${\hat{L}}_{i} = {\hat{α}}_{j k} - {\hat{α}}_{j h}$ 即得到该角的线性化测量方程

\begin{aligned} {\hat{L}}_{i} = & α_{j k}^{0} - α_{j h}^{0} \\ + (a_{j k} - a_{j h}) {\hat{x}}_{j} + (b_{j k} - b_{j h}) {\hat{y}}_{j} \\ - a_{j k} {\hat{x}}_{k} - b_{j k} {\hat{y}}_{k} + a_{j h} {\hat{x}}_{h} + b_{j h} {\hat{y}}_{h} \end{aligned}

角度相关权

考虑在 O 点观测了 3 个独立等精度方向值 $α_{1}, α_{2}, α_{3}$ ，相减得到角度值 $β_{1}, β_{2}$ 。

此时角度值 $β_{1}, β_{2}$ 相关，若视为独立等权，则平差结果不严密。

此例中取 $Q_{α α} = I = diag (1, 1, 1)$ ，有 $β = A α$

[\begin{matrix} β_{1} \\ β_{2} \end{matrix}] = [\begin{matrix} - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix}] [\begin{matrix} α_{1} \\ α_{2} \\ α_{3} \end{matrix}]

故有

Q_{β β} = A^{T} Q_{α α} A = A^{T} A = [\begin{matrix} 2 & - 1 \\ - 1 & 2 \end{matrix}]

P_{β β} = Q_{β β}^{- 1} = \frac{1}{3} [\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}]

同理可求更多角组合的情况。中点多边形的中点也可采用此方法。

9 间接平差的应用 ​

测边网间接平差 ​

单测边误差方程 ​

单未知点 ​

双未知点 ​

测边网求解 ​

精度评定 ​

方向观测网 ​

测角网 ​

单角线性化 ​

角度相关权 ​