# ILRS Reference Point Determination Using Close Range Photogrammetry
## 2 Mathematical Background
### Reference Point Determination  
Satellite laser ranging is a space geodetic technique organized under the International Laser Ranging Service [23]. An SLR telescope is swived-mounted around two axes, the azimuth axis and the elevation axis, and measures the distances to satellites [4]. For this purpose, the telescope transmits a short laser pulse signal towards a satellite that is equipped with retro reflectors, cf. Figure 1. The reflected signal is received by the telescope. The distance \(d\) between the retro reflector and the reference point of the SLR telescope is derived by the observed traveling time \(\Delta t\) and the speed of light \(c_{0}\), i.e.,  
\[d=\frac{\Delta t}{2}c_{0}.\]  
For precise distance measurements, further corrections are applied to compensate for systematic errors, e.g., atmospheric errors or eccentricities of the telescope and satellite constructions ([24] p. 34 ff). The ILRS reference point of an SLR telescope is defined as the intersection point of the telescope axes. If both axes do not intersect, the point onto the azimuth axis, which is closest to the elevation axis, is the reference point [12]. Such a defined point is independent of the telescope orientation and is often referred to as invariant reference point (IRP). If this IRP is known w.r.t. to a reference point of a further space geodetic technique, both techniques become combinable within the intertechnique combination. It is rather an exception that the IRP has been materialized and is observable by means of direct measurements. Usually, mounted markers at the turnable part of the telescope are observed in several telescope orientations. Based on the observed trajectories of these points, the IRP can be derived by means of geometric or transformation approaches. Geometric approaches derive the reference point by separating the trajectories into geometric primitives like spheres, circles, or tori. These approaches are simple and intuitive but have some disadvantages, e.g., a limited number of parameters or metrological restrictions. Transformation approaches overcome these disadvantages and determine the reference  
Figure 1: Schematic representation of the satellite laser ranging (SLR) measurement principle. point by parameterizing the resulting trajectories of the mounted markers. The basic model reads [25]  
\[\mathbf{P}_{i,k}=\mathbf{P}_{\text{IRP}}+\mathbf{R}_{\text{x}}(\beta)\mathbf{R}_{ \text{y}}(\alpha)\mathbf{R}_{\text{z}}^{\text{T}}(\kappa_{k})\mathbf{R}_{\text{ y}}(\gamma)(\mathbf{E}_{\text{AO}}+\mathbf{R}_{\text{x}}(\omega_{k})\mathbf{p}_{i}), \tag{18}\]  
and describes the functional relation of the point \(\mathbf{p}_{i}^{\text{T}}=\left( \begin{array}{cc}x_{i}&y_{i}&z_{i}\\ \end{array} \right)\) in a telescope fixed frame and its corresponding position \(\mathbf{P}_{i,k}^{\text{T}}=\left( \begin{array}{cc}X_{i}&Y_{i}&Z_{i}\\ \end{array} \right)\) in an Earth-fixed frame, e.g., the station network. Matrices \(\mathbf{R}\) are basic rotations around the subindexed coordinate axis using the braced angle. The index \(k\) denotes the orientation of the telescope at the measurement time. The angles \(\omega_{k}\) and \(\kappa_{k}\) are the azimuth angle and the elevation angle, respectively. The axis offset is given by \(\mathbf{E}_{\text{AO}}^{\text{T}}=\left( \begin{array}{cc}0&e_{\text{AO}} &0\\ \end{array} \right)\). The angles \(\alpha\), \(\beta\) parametrize the tilt of the azimuth axis w.r.t. the \(Z\)-axis of the Earth-fixed frame, and \(\gamma\) compensates for the deviation from the orthogonality of the two telescope axes. The translation vector \(\mathbf{P}_{\text{IRP}}\) corresponds to the IRP. Figure 2 depicts a schematic representation of the IRP transformation model. The observations in Equation (18) are the object coordinates \(\mathbf{P}_{i,k}\) that define the trajectories of \(\mathbf{p}_{i}\). The parameters of the right-hand side are parameters to be estimated.