# The Influence of Pillar Displacements on Geodetic Measurements
## 1 Introduction
### Law of Propagation of Variances and Covariances  
There are two ways to calculate the precision of estimates of the calculated quantity. If we have redundant measurements, we can calculate the most likely value of the desired quantity and adjust its precision. If we have just enough measurements, we can calculate the desired quantity monotonically, and then estimate its precision by applying the law of propagation of variances and covariances [12]. In the following, we will consider all the parameters to be random variables. We will focus on the expected value of the parameter described by a random variable \(X_{i}\) and its standard deviation \(\sigma_{X_{i}}\), which reveals the dispersity of the parameter. The expected value \(\mu_{X_{i}}\) of a random variable \(X_{i}\) is evaluated using the following equation [11]:  
\[\mu_{X_{i}}=\mathrm{E}[X_{i}]=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{ \infty}x\mathbf{if}_{\mathbf{X}}(x_{1},x_{2},\ldots,x_{n})\ dx_{1}\cdots\ dx_{n}, \tag{8}\]  
where \(\mathbf{f}_{\mathbf{X}}(x_{1},x_{2},\ldots,x_{n})\) denotes the probability density function of the random vector \(\mathbf{X}\) consisting of all random variables of the problem. With respect to the primary axioms of probability, \(\mathbf{f}_{\mathbf{X}}\) is strictly positive and its integral is equal to one. The variance \(\sigma_{X_{i}}^{2}\) of a random variable \(X_{i}\) is calculated using [11]:  
\[\sigma_{X_{i}}^{2}=\mathrm{E}\Big{[}\big{(}X_{i}-\mu_{X_{i}}\big{)}^{2}\Big{]} =\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\!\!\big{(}x_{i}-\mu_{x_{ i}}\big{)}^{2}\mathbf{f}_{\mathbf{X}}(x_{1},x_{2},\ldots,x_{n})dx_{1}\cdots\ dx_{n}, \tag{9}\]  
while the standard deviation defined as:  
\[\sigma_{X_{i}}=\sqrt{\sigma_{X_{i}}^{2}}. \tag{10}\]  
For any two components of a random vector, we can also calculate the covariance, which describes the statistical correlation between two random variables. The covariance is calculated as follows [11]:  
\[\sigma_{X,X_{i}}=\mathrm{E}\Big{[}\big{(}X_{i}-\mu_{X_{i}}\big{)}\big{(}X_{j}- \mu_{X_{i}}\big{)}\Big{]}=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \!\!(x_{i}-\mu_{x_{i}})\Big{(}\mathbf{x}_{j}-\mu_{x_{j}}\Big{)}\mathbf{f}_{\mathbf{X}}(x_{1},x_{2},\ldots,x_{n})dx_{1}\cdots dx_{n}, \tag{11}\]while the variance is always positive, the covariance can be negative. For any random vector of size \(n\), we can calculate the variances for each component using Equation (9) and the covariances for each pair of components according to Equation (11). The results can be written in the form called variance-covariance matrix [11]:  
\[\mathbf{\Sigma}_{\mathbf{XX}}=\mathrm{E}\big{[}(\mathbf{X}-\mathbf{\mu}_{\mathbf{X}})(\mathbf{X}-\mathbf{ \mu}_{\mathbf{X}})^{\mathrm{T}}\big{]}=\left[ \begin{array}{ccc}\sigma_{X_{1}}^{2}& \cdots&\sigma_{X_{1}X_{n}}\\ \vdots&\ddots&\vdots\\ \sigma_{X_{n}X_{1}}&\cdots&\sigma_{X_{n}}^{2}\end{array} \right]. \tag{12}\]  
Note that \(\mathbf{\Sigma}_{\mathbf{XX}}\) is a symmetrical, since \(\sigma_{X_{i}X_{j}}=\sigma_{X_{j}X_{i}}\), where \(i,\ j=1,\ \ldots,\ n\). Starting from a simple linear transformation of a random vector \(\mathbf{Y}=\mathbf{J}_{\mathbf{0}}+\mathbf{J}\cdot\mathbf{X}\), where \(\mathbf{J}_{\mathbf{0}}\) is a vector of constants and \(\mathbf{J}\) the constant transformation matrix, we can find the following relation:  
\[\mathbf{\Sigma}_{\mathbf{YY}}=\mathbf{J}\cdot\mathbf{\Sigma}_{\mathbf{XX}}\mathbf{J}^{\mathrm{ T}}, \tag{13}\]  
where \(\mathbf{\Sigma}_{\mathbf{YY}}\) is a variance-covariance matrix of a vector. Equation (13) is called the law of propagation of variances and covariances between the random vectors \(\mathbf{X}\) and \(\mathbf{Y}\). The Influence of the Temperature Displacements at Orientation Point \(B\) on the Position of the Control Point \(C\)  
If the temperature load displaces the position of the orientation point \(B\), we use the point \(B^{\prime}(y_{B^{\prime}},x_{B^{\prime}})\), instead of the point \(B(y_{B},x_{B})\) for our measurements.