# The Influence of Pillar Displacements on Geodetic Measurements
## 1 Introduction
### Law of Propagation of Variances and Covariances
Consequently, we measure the horizontal angle \(\alpha^{\prime}=\alpha+\delta_{\alpha}\), where \(\delta_{\alpha}\) is the error of the horizontal angle \(\alpha\), instead of measuring horizontal angle \(\alpha\). We calculate the horizontal angle \(\alpha^{\prime}\) as the difference between the measured horizontal direction to the control point \(C\) and the measured horizontal direction to the orientation point \(B\)[4]:  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]4  
Due to the displaced survey point \(A\), we measure the horizontal distance \(d_{A^{\prime}C}=d_{AC}+\delta_{d_{AC}}\), instead of the horizontal distance \(d_{AC}\), where \(\delta_{d_{AC}}\) represents the error in the horizontal distance \(d_{AC}\). The displacement of the survey point \(A\) leads to an error in the horizontal angle and in the horizontal distance between the points. Assuming that \(\delta_{y_{A}}\) and \(\delta_{x_{A}}\) are small, we can use the following approximation:  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]5  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]6  
where \(\
u_{A}^{B}=\text{atan}\frac{y_{B}-y_{A}}{x_{B}-x_{A}}\) is the displaced bearing between displaced survey point \(A\) and the orientation point \(B\). The non-linear functions relating the coordinates now read:  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]7  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]8  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]9 Displacement at survey point \(A\): geometry. It is also interesting to express the values of the semi-axis of the error ellipse and the bearing of the longer semi-axis of the ellipse, taking into account the same assumptions \(\sigma_{y_{A}}^{2}=\sigma_{X_{A}}^{2}=\sigma_{A}^{2}\) and \(\sigma_{xy_{A}}=0\)[13]. They are as follows:  
\[ \begin{array}{l}a^{2}=\sigma_{A}^{2}\cdot[1-\frac{d_{AC}}{d_{AB}^{2}}\cdot \left(\Delta x_{A}^{B}\cdot\cos\left(\
u_{A}^{B}+\alpha^{\prime}\right)+\Delta y _{A}^{B}\cdot\sin\left(\
u_{A}^{B}+\alpha^{\prime}\right)\right)+\frac{d_{AC}^ {2}}{2d_{AB}^{2}}+\\ \frac{d_{AC}}{d_{AB}}\sqrt{1-\frac{d_{AC}}{d_{AB}^{2}}\cdot\left(\Delta x_{A}^ {B}\cdot\cos\left(\
u_{A}^{B}+\alpha^{\prime}\right)+\Delta y_{A}^{B}\cdot \sin\left(\
u_{A}^{B}+\alpha^{\prime}\right)\right)+\frac{d_{AC}^{2}}{4d_{AB}^ {2}}}],\\ \\ b^{2}=\sigma_{A}^{2}\cdot[1-\frac{d_{AC}}{d_{AB}^{2}}\cdot\left(\Delta x_{A}^ {B}\cdot\cos\left(\
u_{A}^{B}+\alpha^{\prime}\right)+\Delta y_{A}^{B}\cdot \sin\left(\
u_{A}^{B}+\alpha^{\prime}\right)\right)+\frac{d_{AC}^{2}}{2d_{AB}^ {2}}-\\ \frac{d_{AC}}{d_{AB}}\sqrt{1-\frac{d_{AC}}{d_{AB}^{2}}\cdot\left(\Delta x_{A}^ {B}\cdot\cos\left(\
u_{A}^{B}+\alpha^{\prime}\right)+\Delta y_{A}^{B}\cdot \sin\left(\
u_{A}^{B}+\alpha^{\prime}\right)\right)+\frac{d_{AC}^{2}}{4d_{AB}^ {2}}}],\\ \\ \tan 2\theta=\frac{\Delta x_{A}^{B}\cdot\sin\left(\
u_{A}^{B}+\alpha^{\prime} \right)+\Delta y_{A}^{B}\cdot\cos\left(\
u_{A}^{B}+\alpha^{\prime}\right)-d_{ AC}/2\cdot\sin\left(2*\left(\
u_{A}^{B}+\alpha^{\prime}\right)\right)}{ \Delta x_{A}^{B}\cos\left(\
u_{A}^{B}+\alpha^{\prime}\right)-\Delta y_{A}^{B} \cdot\sin\left(\
u_{A}^{B}+\alpha^{\prime}\right)-d_{AC}/2\cdot\cos\left(2* \left(\
u_{A}^{B}+\alpha^{\prime}\right)\right)}.\end{array}  \tag{54}\]  
For numerical values, the reader is referred to Section 3.3.