## 2 Materials and Methods
### Experimental Works  
The experimental works were performed in Wroclaw, Poland. The proposed algorithms were tested on data obtained during measurements at the points of the spatial traverse (Figure 2). At the total station traverse positions 1, 2, 5, there were spatial distances \(s\), horizontal and vertical angles \(\alpha\), \(\beta\) (Figure 1), surveyed with standard deviations \(\sigma_{d}\) = 0.006 m, \(\sigma_{\alpha}\) = \(\sigma_{\beta}\) = 0.0010 grad, and the instrument height \(i\) and reflector (prism) height \(j\), with \(\sigma_{i}\) = \(\sigma_{j}\) = 0.002 m. Measurements were performed with the use of the Leica FlexLine TS02 total station [36]. The \(X\), \(Y\), \(Z\) coordinates of the station points 1, 2, 5 and merging points 3, 4, 6, 7 were measured with a Leica GS10 GNSS receiver [37] with standard deviations \(\sigma_{X}\) = \(\sigma_{Y}\) = \(\sigma_{Z}\) = 0.008 m. At each point, we performed real-time kinematic (RTK) GNSS surveys with direct differential corrections broadcast by the GNSS permanent station WROC, located at the Wroclaw University of Environmental and Life Sciences. The observation data can be downloaded from [[http://www.asegupos.pl/index.php?wrg_type=syst_descr&sub=ref_st&st_id=wroc](http://www.asegupos.pl/index.php?wrg_type=syst_descr&sub=ref_st&st_id=wroc)]([http://www.asegupos.pl/index.php?wrg_type=syst_descr&sub=ref_st&st_id=wroc](http://www.asegupos.pl/index.php?wrg_type=syst_descr&sub=ref_st&st_id=wroc)) (accessed on 18 July 2023). The distance from the mobile receiver to the reference station WROC was less than 250 m.  
The northern \(\xi\) and eastern \(\eta\) components of the deflection of the total station vertical axis from normal to the GRS80 ellipsoid were computed at the points 1, 2, 5 from the EGM2008 geoid model data, available from [[https://www.usna.edu/Users/oceano/pguth/md_help/html/egm96.htm](https://www.usna.edu/Users/oceano/pguth/md_help/html/egm96.htm)]([https://www.usna.edu/Users/oceano/pguth/md_help/html/egm96.htm](https://www.usna.edu/Users/oceano/pguth/md_help/html/egm96.htm)) (accessed on 18 July 2023) and presented in Table 1.  
The approximate value \(\Sigma_{1}\) was computed by solving Equation (4) for GNSS-surveyed point coordinates \(X,Y,Z\) of the points 1 and 2, and \(x,y,z\) coordinates of the point 2 measured using total station set up in point 1. Analogically, we computed the values \(\Sigma_{2}\) and \(\Sigma_{5}\). According to information from total station manufacturers, the direction of the vertical axis of an instrument is consistent with the real plumb line direction to the order of 0.5\({}^{\prime\prime}\)-2.0\({}^{\prime\prime}\) [arc seconds]. So, in the adjustment process, the standard deviations of the components of deflection of the vertical are assumed to be 1\({}^{\prime\prime}\) (\(\sigma_{L}=\sigma_{\eta}\) = 0.0003 [grad]). The covariance \(\sigma_{\varepsilon\eta}\) is unknown; it is assumed as 0 in the matrix (31). Research has shown [38] that in mountainous areas, in the Alps, the actual direction of the vertical can differ up to 3.5\({}^{\prime\prime}\) from the direction of the vertical computed in the EGM2008 model. This error is similar to the accuracy of the vertical angles used in the experiment's total-station instrument (3.0\({}^{\prime\prime}\)). In the case of unique high-accuracy measurements using total stations in mountain areas, the accuracy of the vertical deflection computed from EGM2008 model can be improved up to 0.8\({}^{\prime\prime}\) using a correction computed from the numerical terrain model [38]. An alternative method of joint total station and GPS positioning with the use of digital terrain and gravity models is given in the paper [39]. In this case, the spherical model of the local vertical distribution is extended using local digital models of the terrain and gravity. Then, the residual field of the vertical deflection, based on spherical harmonic polynomials' expansion, is determined via real-time adjustment of the total station and GPS data.