# Sphero-Conical Modeling for the Estimation of Very Long Baseline Interferometry Invariant Point
## 6 Conclusions  
In this study, we proposed a sphero-conical model and successfully applied it to estimate the VLBI IVP, which is the primary concern in establishing and maintaining global geodetic reference frames based on several space-geodetic techniques. It is very important to note that only three unknowns are necessary to estimate the IVP (and one additional  
Figure 8: Residuals of each observation after TLS adjustment. The \(z\)-component is not involved in the azimuthal rotation, and the periodic large residuals are related to the vertical motion at specific azimuth. Figure 9: Correlation between parameters after TLS adjustment. All correlations are resolved except the one between the UP component (\(\gamma\)) and the slant heights of the vertical targets. parameter per target). In other words, all measured target positions are directly connected to the IVP; thus, no single target is in vain during several steps of 3D circle fitting, as in conventional methods. The total number of parameters is significantly reduced in the proposed method, especially for the traces of vertical targets (e.g., only 3% of parameters are necessary compared to the conventional 3D circle fitting algorithm). Thus, this model can increase the separability between parameters and the numerical stability of the system. More importantly, the newly developed sphero-conical model can overcome the limitation of geometry arising from the VLBI antenna rotation. Especially, the conventional model should estimate the 3D circle based on partial data only (e.g., a quarter-circle or less), resulting in considerable instability in the estimation process. As explained earlier, ground survey measurements, such as slant distances and horizontal/vertical angles, can be directly used in a sphero-conical model. In addition, when combined with TLS within the GHM, we can estimate the errors in all components of the target coordinates in the 3D space. It should be noted that the vertical offset of the antenna axis must be considered, although it was assumed to be parallel to the plumbline in this study. There might be two types of offsets: (a) the horizontal offset of the vertical axis or (b) the inclination of the vertical axis. These offsets should be modeled to complete the estimation of the invariant point [17]. Because several types of ground survey measurements with possibly different levels of precision are involved in the adjustment, the variance component model needs to be considered instead of a single variance component [27; 34; 35; 36]. Lastly, two motions (azimuthal rotation and motion in elevation) were modeled differently in this study: either the radii of the horizontal circles in the 3D space or the slant distances of the cones with the apex at IVP. However, a unified, homogeneous model that incorporates two motions makes the system simpler and more consistent, which will be discussed in future research. Conceptualization, T-S.B. and C.-K.H. ; methodology, T-S.B. ; software, T-S.B. ; validation, T-S.B. and C.-K.H. ; formal analysis, T-S.B. and C.-K.H. ; investigation, T-S.B. and C.-K.H. ; resources, T-S.B. and C.-K.H. ; data curation, T-S.B. ; writing--original draft preparation, T-S.B. and C.-K.H. ; writing--review and editing, T-S.B. and C.-K.H. ; visualization, T-S.B. ; supervision, T-S.B. ; project administration, T-S.B. ; funding acquisition, T-S.B. All authors have read and agreed to the published version of the manuscript. This research was supported by a grant from the National R&D Project \"Development of ground-based centimeter-level maritime precise PNT technologies\" funded by the Ministry of Oceans and Fisheries (1525012253). Not applicable. The authors would like to thank the National Geographic Information Institute (NGII) of Korea for providing the observation data used in this experiment. The authors declare no conflict of interest.