## 2 Materials and Methods
### Functional Adjustment Model  
The creation of the functional model of the adjustment of a geodetic network involves the listing of observation equations and transforming them into linear equations of correction. The formulas below are general equations for three types of observations (in an integrated spatial geodetic network): (1) the station-target distance; (2) the left target-right target distance; (3) components of the GNSS vector. (1) The station-target spatial distance (cf. Equation (5)):  
\[d_{jk}+v_{jk}^{(d)}=\sqrt{\left(x_{k}-x_{j}\right)^{2}+\left(y_{k}-y_{j} \right)^{2}+\left(z_{k}-z_{j}\right)^{2}} \tag{11}\]  
\[v_{jk}^{(d)}=\frac{\partial d_{jk}}{\partial x_{j}}\cdot\delta x_{j}+\frac{ \partial d_{jk}}{\partial y_{j}}\cdot\delta y_{j}+\frac{\partial d_{jk}}{ \partial z_{j}}\cdot\delta z_{j}+\frac{\partial d_{jk}}{\partial x_{k}}\cdot \delta x_{k}+\frac{\partial d_{jk}}{\partial y_{k}}\cdot\delta y_{k}+\frac{ \partial d_{jk}}{\partial z_{k}}\cdot\delta z_{k}+l_{jk}^{(d)} \tag{12}\]  
\[l_{jk}^{(d)}=d_{jk}^{(0)}-d_{jk} \tag{13}\]  
where:  
\(l_{jk}^{(d)}\)--the absolute term in the correction Equation (12);  
\[d_{jk}^{(0)}=\sqrt{\left(x_{k}^{(0)}-x_{j}^{(0)}\right)^{2}+\left(y_{k}^{(0)}-y _{j}^{(0)}\right)^{2}+\left(z_{k}^{(0)}-z_{j}^{(0)}\right)^{2}}\]  
--the approximate distance;\[\left\{ \begin{array}{ll}\frac{\partial d_{jk}}{\partial x_{j}}=-\frac{x_{k}^{(0)} -x_{j}^{(0)}}{d_{jk}^{(0)}};&\frac{\partial d_{jk}}{\partial y_{j}}=-\frac{y_{k }^{(0)}-y_{j}^{(0)}}{d_{jk}^{(0)}};&\frac{\partial d_{jk}}{\partial z_{j}}=- \frac{z_{k}^{(0)}-z_{j}^{(0)}}{d_{jk}^{(0)}}\\ \frac{\partial d_{jk}}{\partial x_{k}}=\frac{x_{k}^{(0)}-x_{j}^{(0)}}{d_{jk}^{(0 )}};&\frac{\partial d_{jk}}{\partial y_{k}}=\frac{y_{k}^{(0)}-y_{j}^{(0)}}{d_{ jk}^{(0)}};&\frac{\partial d_{jk}}{\partial z_{k}}=\frac{z_{k}^{(0)}-z_{j}^{(0)}}{d_{jk}^{(0 )}}\\ x^{(0)}\ y^{(0)}\ z^{(0)}\ \mbox{--approximate coordinates.}\\ \end{array} \right.\]  
\(\left. \begin{array}{ll}\mbox{(2) The left target--right target spatial distance (cf. Equation (8)):}\\ \end{array} \right.\)  
\[d_{LR}+v_{LR}^{(d)}=\sqrt{\left(x_{R}-x_{L}\right)^{2}+\left(y_{R}-y_{L} \right)^{2}+\left(z_{R}-z_{L}\right)^{2}} \tag{14}\]  
\[v_{LR}^{(d)}=\frac{\partial d_{LR}}{\partial x_{L}}\cdot\delta x_{L}+\frac{ \partial d_{LR}}{\partial y_{L}}\cdot\delta y_{L}+\frac{\partial d_{LR}}{ \partial z_{L}}\cdot\delta z_{L}+\frac{\partial d_{LR}}{\partial x_{R}}\cdot \delta x_{R}+\frac{\partial d_{LR}}{\partial y_{R}}\cdot\delta y_{R}+\frac{ \partial d_{LR}}{\partial z_{R}}\cdot\delta z_{R}+l_{LR}^{(d)} \tag{15}\]  
\[l_{LR}^{(d)}=d_{LR}^{(0)}-d_{LR} \tag{16}\]  
where:  
\(l_{LR}^{(d)}\)--the absolute term in the correction Equation (15):  
\[d_{LR}^{(0)}=\sqrt{\left(x_{R}^{(0)}-x_{L}^{(0)}\right)^{2}+\left(y_{R}^{(0)} -y_{L}^{(0)}\right)^{2}+\left(z_{R}^{(0)}-z_{L}^{(0)}\right)^{2}}\mbox{--the approximate distance.}\]  
The partial derivatives in Equation (15) are calculated similarly as for Equation (12). (3) The GNSS vector (\(\Delta x\), \(\Delta y\), \(\Delta z\)) between two points \(j\) and \(k\):  
\[\left\{ \begin{array}{ll}\Delta x_{jk}+v_{jk}^{(\Delta x)}=x_{k}-x_{j}\\ \Delta y_{jk}+v_{jk}^{(\Delta y)}=y_{k}-y_{j}\\ \Delta z_{jk}+v_{jk}^{(\Delta z)}=z_{k}-z_{j}\end{array} \right. \tag{17}\]  
\[\left\{ \begin{array}{ll}v_{jk}^{(\Delta x)}=\frac{\partial(\Delta x)_{jk}}{ \partial x_{k}}\cdot\delta x_{k}-\frac{\partial(\Delta x)_{jk}}{\partial x_{ j}}\cdot\delta x_{j}+l_{jk}^{(\Delta x)}=\delta x_{k}-\delta x_{j}+l_{jk}^{( \Delta x)}\\ v_{jk}^{(\Delta y)}=\frac{\partial(\Delta y)_{jk}}{\partial y_{k}}\cdot \delta y_{k}-\frac{\partial(\Delta y)_{jk}}{\partial x_{j}}\cdot\delta y_{j}+ l_{jk}^{(\Delta y)}=\delta y_{k}-\delta y_{j}+l_{jk}^{(\Delta y)}\\ v_{jk}^{(\Delta z)}=\frac{\partial(\Delta z)_{jk}}{\partial z_{k}}\cdot \delta z_{k}-\frac{\partial(\Delta z)_{jk}}{\partial z_{j}}\cdot\delta z_{j}+ l_{jk}^{(\Delta z)}=\delta z_{k}-\delta z_{j}+l_{jk}^{(\Delta z)}\\ \end{array} \right. \tag{18}\]  
\[\left\{