# The Impact of Gravity on Different Height Systems: A Case Study on Mt. Medvednica
## Appendix A Appendix A
### Calculation of the Gravimeter Drift  
To explain the procedure for calculating the gravimeter drift, the following section provides a numerical example of gravimeter drift calculation for the first day of measurement. The values of gravity after corrections for all the points measured on the first day, along with the corresponding measurement times in UTC, are given in Table 14.  
The first step in calculating the gravimeter drift is to extract all the measurements of the repeatedly measured point, based on which the gravimeter drift will be calculated (in this case, AGT03-E3), along with the time of those measurements in seconds.  
Based on the data from Table 15, the design matrix and the observation vector are created, which will be used for the adjustment to obtain the coefficients of the line equation.  
\[\mathbf{A}=\begin{bmatrix}23704&1\\ 23764&1\\ 37872&1\\ 37932&1\end{bmatrix}\qquad I=\begin{bmatrix}4343.73319\\ 4343.73495\\ 4343.81048\\ 4343.81093\end{bmatrix}\]  
After the matrix \(\mathbf{A}\) and vector \(I\) have been created, the adjustment is performed using the least squares method, applying the following expressions:  
\[\mathbf{N}=\mathbf{A}^{T}\mathbf{A}=\begin{bmatrix}3999732320&123272\\ 123272&4\end{bmatrix}\]  
\[\mathbf{n}=\mathbf{A}^{T}I=\begin{bmatrix}535466595.6\\ 17375.08955\end{bmatrix}\]  
\[\mathbf{Q}=\mathbf{N}^{-1}=\begin{bmatrix}4.98167\text{E}-09&-0.000153525\\ -0.000153525&4.981338458\end{bmatrix}\]  
\[\mathbf{x}=\mathbf{Q}\mathbf{n}=\begin{bmatrix}0.000005\\ 4343.60569\end{bmatrix}\]  
In this way, the equation of the line for the gravimeter drift model is obtained, which best describes all the measurements of point AGT03-E3 during the day. The upper element  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Point** & **Gravity [mGal]** & **Time (UTC) [s]** \\ \hline AGT03-E3 & 4343.73319 & 23704 \\ AGT03-E3 & 4343.73495 & 23764 \\ AGT03-E3 & 4343.73401 & 37872 \\ AGT03-E3 & 4343.73413 & 37932 \\ \hline \hline \end{tabular}
\end{table}
Table 14: The values of the gravity for all the measurements on the first day, corrected for all the corrections except the gravimeter drift, along with the corresponding measurement times.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Point** & **Gravity [mGal]** & **Time (UTC) [s]** \\ \hline AGT03-E3 & 4343.73319 & 23704 \\ AGT03-E3 & 4343.73495 & 23764 \\ AGT03-E3 & 4343.73401 & 37872 \\ AGT03-E3 & 4343.73413 & 37932 \\ \hline \hline \end{tabular}
\end{table}
Table 15: The extracted values of the acceleration of gravity and the measurement times for the point based on which the drift will be calculated.  
of the vector \(x\) represents the coefficient \(a\), while the lower element represents the coefficient \(b\), so the equation of the line is as follows:  
\[y=0.000005x+4343.60569\]  
If the values of the seconds corresponding to the last and first measurements from Table A3 are substituted into the equation of the line in place of x, and these two resulting values are subtracted, the total gravimeter drift in mGal between the first and last measurement of point AGT03-E3 during the day will be obtained:  
\[g_{d}=(0.000005\cdot 37932+4343.60569)-(0.000005\cdot 23704+4343.60569)g_{d}=0.07696 \ mGal\]  
Finally, the value of the correction for the gravimeter drift for every point in a day will be obtained using the following expression:  
\[g_{di}=g_{d}\frac{t_{i}-23704}{37932-23704}\]  
where \(t_{i}\) represents the measurement time in seconds for each point during the day from Table A2. The final values of the gravimeter drift for all points during the first day are as follows: