# Sphero-Conical Modeling for the Estimation of Very Long Baseline Interferometry Invariant Point
###### Abstract  
A geodetic reference frame is a fundamental element in geoinformation fields such as autonomous navigation and digital twins. The international terrestrial reference frame is established and maintained using several space-geodetic techniques, including very long baseline interferometry (VLBI) and satellite laser ranging (SLR). For several decades, geodesists have been devoted to connecting these two sensors at a site (local tie). The reference point of the VLBI antenna and SLR telescope, called invariant point (IVP), should be precisely determined to connect these two solutions. We developed an innovative integrated model to estimate the IVP, which is composed of spherical and conical models, depending on the rotational axis. In this model, all target points in 3D spaces were directly connected to the IVP; thus, the stability and robustness of the system were secured. Furthermore, all inherent errors in the coordinates were predicted by applying the total least-squares approach. very long baseline interferometry; invariant point; international terrestrial reference frame; local tie; total least-squares method 20  
The coordinates of the targets, together with their variance-covariance information, were obtained from ground surveys by applying the least-squares method. Then, the IVP estimation procedure is performed using the geometric relationship between the targets and IVP. The 3D circle fitting method is commonly used to estimate the IVP. The trace of a target on the antenna rotating about an independent axis can be expressed as a 3D circle that can be defined using seven parameters, including three parameters for the center of the circle, three parameters for the unit normal vector, and one parameter for the radius of the circle [8; 10; 11; 12; 13; 14]. Slightly different approaches have also been introduced by imposing various types of geometric conditions [5; 15]. The trace of each target due to the horizontal rotation presents a full circle in normal surveying conditions, which increases the stability for parameter estimation. However, the trace of a target by the vertical motion in elevation makes only a quarter circle because of the limited motion of the rotational axis. Consequently, it leads to an unstable linear system being solved. Moreover, each rotational sequence of the elevation axis yields seven parameters for modeling each arc. The increase of rotational sequences requires more parameters to be estimated, which is one of the disadvantages of the 3D circle fitting method. In this study, we propose a new model for the determination of the IVP from the coordinates of targets estimated based on terrestrial observations. The basic concept of the new approach is that the trace of horizontal (azimuthal) rotation becomes a 3D circle in space. However, the motion in elevation creates a cone shape with an apex at the IVP, where the slant distance of the cone is the only unknown for a vertical target. Therefore, a new approach, called the sphero-conical model, connects all target positions in the 3D space to a common IVP, which significantly reduces the unknowns compared to the conventional methods. In other words, the primary unknowns are the coordinates of the IVP and an additional parameter for each target, representing either horizontal or vertical motions. Setting aside the small number of observations, the conventional model based on a 3D circle may lead to an ill-conditioned linear system when circle arcs are incomplete; as a consequence, the quality of the estimated parameters will be degraded [5]. However, the new sphero-conical model can overcome the ill-conditioned problem and obtain efficiency in terms of the redundancy of the linear system to be solved, which is the most significant contribution of this study. It should be noted that the coordinates of the observed targets are observations; hence, they have inherent random errors. Therefore, the adjustment computation must be performed by estimating not only the IVP, but also the coordinates of the targets on the antenna.