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+# 10 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
+
+![](./images/sigle-unknown-segment.svg)
+
+- 参数：坐标 $X_k,Y_k$
+- 近似值：坐标 $X_k^0,Y_k^0$，边长 $L_{jk}^0$
+
+有观测方程
+
+$$
+\hat L_{jk}=\sqrt{(\hat X_k-X_j)^2+(\hat Y_k-Y_j)^2}
+$$
+
+记 $\hat X_k=X_k^0+\hat x_k$，$\hat Y_k=Y_k^0+\hat y_k$，有线性化
+
+$$
+\begin{aligned}
+\hat L_{jk}
+&=L_{jk}^0+\frac{\partial \hat L_{jk}}{\partial\hat X_k}\hat x_k
+  +\frac{\partial \hat L_{jk}}{\partial\hat Y_k}\hat y_k \\
+&=L_{jk}^0+\frac{X_k^0-X_j}{L_{jk}^0}\hat x_k
+  +\frac{Y_k^0-Y_j}{L_{jk}^0}\hat y_k \\
+&=L_{jk}^0+\hat x_k\cos\alpha_{jk}^0+\hat y_k\sin\alpha_{jk}^0
+\end{aligned}
+$$
+
+#### 双未知点
+
+![](./images/double-unknown-segment.svg)
+
+- 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 $L_{jk}^0$
+
+有观测方程
+
+$$
+\hat L_{jk}=\sqrt{(\hat X_k-\hat X_j)^2+(\hat Y_k-\hat Y_j)^2}
+$$
+
+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
+
+$$
+\begin{aligned}
+\hat L_{jk}
+&=L_{jk}^0+\frac{\partial \hat L_{jk}}{\partial X_j}\hat x_j
+  +\frac{\partial \hat L_{jk}}{\partial\hat Y_j}\hat y_j
+  +\frac{\partial \hat L_{jk}}{\partial\hat X_k}\hat x_k
+  +\frac{\partial \hat L_{jk}}{\partial\hat Y_k}\hat y_k \\
+&=L_{jk}^0+\frac{X_k^0-X_j^0}{L_{jk}^0}(\hat x_k-\hat x_j)
+  +\frac{Y_k^0-Y_j^0}{L_{jk}^0}(\hat y_k-\hat y_j) \\
+&=L_{jk}^0+(\hat x_k-\hat x_j)\cos\alpha_{jk}^0+(\hat x_k-\hat x_j)\sin\alpha_{jk}^0
+\end{aligned}
+$$
+
+### 测边网求解
+
+![](./images/edge-net.svg)
+
+$$
+\begin{aligned}
+{\color{blue}\hat L_1}
+&=\sqrt{(X_A-{\color{red}\hat X_2})^2
++(Y_A-{\color{red}\hat Y_2})^2} \\
+
+{\color{blue}\hat L_2}
+&=\sqrt{(X_B-{\color{red}\hat X_2})^2
++(Y_B-{\color{red}\hat Y_2})^2} \\
+
+{\color{blue}\hat L_3}
+&=\sqrt{({\color{red}\hat X_1}-{\color{red}\hat X_2})^2
++({\color{red}\hat Y_1}-{\color{red}\hat Y_2})^2} \\
+
+{\color{blue}\hat L_4}
+&=\sqrt{(X_A-{\color{red}\hat X_1})^2
++(Y_A-{\color{red}\hat Y_1})^2} \\
+
+{\color{blue}\hat L_5}
+&=\sqrt{(X_B-{\color{red}\hat X_1})^2
++(Y_B-{\color{red}\hat Y_1})^2} \\
+\end{aligned}
+$$
+
+线性化得到 $\hat{\boldsymbol L}=\boldsymbol B\hat{\boldsymbol X}+\boldsymbol d$
+
+$$
+\begin{bmatrix}
+\hat L_1\\
+\hat L_2\\
+\hat L_3\\
+\hat L_4\\
+\hat L_5
+\end{bmatrix}
+=
+\begin{bmatrix}
+0 & 0 & \cos\alpha^0_{A2} & \sin\alpha^0_{A2}\\
+0 & 0 & \cos\alpha^0_{B2} & \sin\alpha^0_{B2}\\
+\cos\alpha^0_{21} & \sin\alpha^0_{21} & -\cos\alpha^0_{21} & -\sin\alpha^0_{21}\\
+\cos\alpha^0_{A1} & \sin\alpha^0_{A1} & 0 & 0\\
+\cos\alpha^0_{B1} & \sin\alpha^0_{B1} & 0 & 0
+\end{bmatrix}
+\begin{bmatrix}
+\hat x_1\\
+\hat y_1\\
+\hat x_2\\
+\hat y_2
+\end{bmatrix}
++
+\begin{bmatrix}
+L^0_{A2}\\
+L^0_{B2}\\
+L^0_{21}\\
+L^0_{A1}\\
+L^0_{B1}
+\end{bmatrix}
+$$
+
+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
+
+$$
+\hat{\sigma}_0^2
+=
+\frac{\boldsymbol V^{\rm T}\boldsymbol{PV}}{n-t}
+=
+\frac{\boldsymbol V^{\rm T}\boldsymbol{PV}}{r}
+\qquad
+\boldsymbol{Q}_{\hat{x}\hat{x}}=\boldsymbol{N}^{-1}
+$$
+
+- 函数式
+
+  $$
+  \hat{F}
+  =
+  \hat{L}_{jk}
+  =
+  \sqrt{
+  (\hat{X}_k-\hat{X}_j)^2
+  +
+  (\hat{Y}_k-\hat{Y}_j)^2
+  }
+  $$
+
+- 权函数式
+
+  $$
+  \begin{gathered}
+  \mathrm d\hat{F}
+  =
+  -\frac{\Delta X^0_{jk}}{L^0_{jk}}\mathrm d\hat{x}_j
+  -\frac{\Delta Y^0_{jk}}{L^0_{jk}}\mathrm d\hat{y}_j
+  +\frac{\Delta X^0_{jk}}{L^0_{jk}}\mathrm d\hat{x}_k
+  +\frac{\Delta Y^0_{jk}}{L^0_{jk}}\mathrm d\hat{y}_k
+  =
+  \boldsymbol{f}^{T}\mathrm d\hat{\boldsymbol{x}} \\
+
+  Q_{\hat{F}\hat{F}}
+  =
+  \boldsymbol{f}^{T}\boldsymbol{Q}_{\hat{x}\hat{x}}\boldsymbol{f} \\
+
+  \hat{\sigma}_{\hat{F}}
+  =
+  \hat{\sigma}_0\sqrt{Q_{\hat{F}\hat{F}}}
+  \end{gathered}
+  $$
+
+- 相对精度
+  $$
+  \frac{\hat{\sigma}_{\hat{F}}}{\hat{F}}
+  =
+  \frac{\hat{\sigma}_0\sqrt{Q_{\hat{F}\hat{F}}}}{\hat{L}_{jk}}
+  $$