# Sphero-Conical Modeling for the Estimation of Very Long Baseline Interferometry Invariant Point
## 5 Experiments and Discussion  
To evaluate the completeness and/or compatibility of the mathematical models, numerical verification was performed using the survey results from March 2019 [33]. The initial coordinates are given in Table 1, along with the IVP estimated using conventional methods (the last row of the table). The local coordinate system was established by using VP01 as the origin, and VP02 was oriented to the \(\mathbf{x}\)-axis (east direction), where height information was adopted from a spirit leveling survey (orthometric height). The estimated IVP values were based on the 3D circle fitting of the conventional LESS for each target, which was averaged to generate the final solution. The observations are the coordinates \((x,y,z)\) of each target in 3D. The total number of observations was 423 points, of which 241 points were related to azimuthal rotations and 182 points were used for the vertical motion model. Figure 6 shows a trace of the azimuthal rotation of each target, where the symbols represent the pillars related to the determination of the coordinates. As shown in the figure, only partial directions are covered by a specific pillar; thus, full coverage (360\({}^{\circ}\) around the antenna) is completed by five pillars. For vertical motion, only two targets were used, that is, #6 and #7, as shown in Figure 7. The vertical motion of the antenna was conducted every 30\({}^{\circ}\) in the azimuth. Table 2 describes the parameters used in the model, where the IVP is expressed using only three parameters. Eleven parameters are related to the radii of the horizontal circles (\(r_{k}\)) by the azimuthal rotation of each target, resulting in the same number of unknowns. Similarly, distance (\(c_{l}\)) explains the slant heights of a cone traced by the vertical motion of two targets with the apex at the IVP. It is remarkable that only five parameters, i.e., the coordinates of the IVP and the slant height of two cones, are used for the trace of vertical targets, regardless of the number of azimuthal sequences. Contrary to the proposed model, a total of 168 parameters (2 targets \(\times\) 7 parameters for 3D circle \(\times\) 12 azimuthal rotation sequences, assuming 30\({}^{\circ}\) interval in azimuth) are required to implement the conventional 3D circle fitting algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Station** & \(\mathbf{x}\)**[m]** & \(\mathbf{y}\)**[m]** & \(\mathbf{z}\)**[m]** \\ \hline VP01 & 0.0000 & 0.0000 & 177.9075 \\ VP02 & 46.5314 & 0.0000 & 180.8358 \\ VP03 & 64.9459 & 15.4701 & 180.9527 \\ VP04 & 38.0156 & 65.5444 & 180.5815 \\ VP05 & 1.0564 & 50.7578 & 185.5912 \\ SEJN & 7.1632 & \(-\)26.3145 & 181.1960 \\ \hline \hline _est_. IVP & 37.6278 & 23.4907 & 194.5977 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial coordinates of each pillar/CORS along with the previous results of IVP estimation. Figure 6: The trace of azimuthal rotation of targets. Each symbol represents the related pillars (P1 to P5) on which the instrument was installed for ground surveying. \begin{table}
\begin{tabular}{c c l} \hline
**Parameter** & **Notation** & **Description** \\ \hline
1–3 & \((\alpha,\beta,\gamma)\) & IVP where \((\alpha^{0},\beta^{0},\gamma^{0})=(37.6,23.5,194.0)\) [m] \\
4–14 & \(r_{k}\) & Radii of horizontal circles \\
15–16 & \(c_{l}\) & Slant heights of a cone traced by vertical motion, with the apex at IVP \\ \hline \end{tabular}
\end{table} 
Table 2: List of parameters used in this experiment. Figure 7: The trace of vertical motion of targets (only two targets are involved). Two targets are located on the opposite sides of the structure under the antenna. to be attributed to either rotation at a specific azimuth or measurements at a specific pillar, which should be investigated further. To verify the stability and separability between the parameters of the system, all correlations between the parameters listed in Table 4 were investigated by calculating cross-correlations (see Figure 9).