## 2 Materials and Methods
### The Procedure for Adjusting the Integrated Network  
The adjustment of the integrated geodetic network (using the method of least squares) will be based on an overdetermined system of equations of correction type (12), (15) and (18), expressed as the following matrix form:  
\[\mathbf{V}=\mathbf{A}\cdot\mathbf{X}-\mathbf{L} \tag{21}\]  
where:  
\(\mathbf{V}=\left[\left\{v_{jk}^{(d)};v_{LR}^{(d)};\left(v_{jk}^{(\Delta x)};v_{ jk}^{(\Delta y)};v_{jk}^{(\Delta z)}\right)\right\}\right]^{\mathsf{T}}\)--the vector of corrections type (12), (15) and (18) to be determined (curly brackets \(\left\{\ \ldots\ \right\}\) stand for all elements of a type);  
**A**--the matrix of coefficients of the unknowns (partial derivatives) in Equations (12), (15) and (18);  
\(\mathbf{X}=\left[\left\{\left(\delta x_{L};\delta y_{L};\delta z_{L};\delta x_{ R};\delta y_{R};\delta z_{R}\right);\left(\delta x_{j};\delta y_{j};\delta z_{j}; \delta x_{k};\delta y_{k};\delta z_{k}\right)\right\}\right]^{\mathsf{T}}\)--the vector of the unknowns--increments to approximate coordinates;  
\(\mathbf{L}=\left[\left\{l_{jk}^{(d)};l_{LR}^{(d)};\left(l_{jk}^{(\Delta x)};l_ {jk}^{(\Delta y)};l_{jk}^{(\Delta z)}\right)\right\}\right]^{\mathsf{T}}\)--the vector of absolute terms type (13), (16) and (19).  
The estimated vector of the unknowns \(\mathbf{\hat{X}}\) is calculated with the method known from the adjustment calculus [13], which stems from the imposition of the least square condition (\(\mathbf{V^{T}}\cdot\mathbf{P}\cdot\mathbf{V}\) = min.) on the system (21):  
\[\mathbf{\hat{X}}=\left(\mathbf{A^{T}}\cdot\mathbf{P}\cdot\mathbf{A}\right)^{ -1}\mathbf{A^{T}}\cdot\mathbf{P}\cdot\mathbf{L} \tag{22}\]  
where:  
**P**--the matrix of weights set up from mean errors of the linear pseudo-observations (9), (10) and mean errors of GNSS vector measurements (\(m^{(\Delta x)};m^{(\Delta y)};m^{(\Delta z)}\)):  
\[diag\{\mathbf{P}\}=\left\{\frac{1}{\left(m_{jk}^{(d)}\right)^{2}};\ \frac{1}{\left(m_{LR}^{(d)}\right)^{2}};\ \left(\frac{1}{\left(m_{jk}^{(\Delta x)}\right)^{2}};\ \frac{1}{\left(m_{jk}^{(\Delta y)}\right)^{2}};\ \frac{1}{\left(m_{jk}^{(\Delta z)} \right)^{2}}\right)\right\} \tag{23}\]  
The next step is to substitute the vector of unknowns \(\mathbf{X}\) (21) with the calculated vector \(\mathbf{\hat{X}}\) (22) and calculate the vector of observation corrections \(\mathbf{V}\), which are used to adjust the observations--the left sides of the observation Equations (11), (14) and (17).  
Information on mean errors of the adjusted coordinates (\(m_{x}\), \(m_{y}\), \(m_{z}\)) can be found on the diagonal of the covariance matrix (\(\mathbf{Q_{x}}\)) of the vector \(\mathbf{X}\):  
\[\mathbf{Q_{x}}=m_{0}\cdot\left(\mathbf{A^{T}}\cdot\mathbf{P}\cdot\mathbf{A} \right)^{-1} \tag{24}\]  
\[m_{0}=\sqrt{\frac{\mathbf{V^{T}}\cdot\mathbf{P}\cdot\mathbf{V}}{r}} \tag{25}\]  
where:  
\(m_{0}\)--the standard error of unit weight;  
\(r\)--the number of redundant observations.  
Next, a single parameter characterising the point's accuracy is calculated for each point--the error of position (\(m_{P}\)) in a three-dimensional Cartesian system:  
\[m_{P}=\sqrt{m_{x}^{2}+m_{y}^{2}+m_{z}^{2}} \tag{26}\]