# ILRS Reference Point Determination Using Close Range Photogrammetry
## 3 Satellite Observing System Wettzell
### Analysis and Results
#### 3.2.3 Bias of the Estimates  
The functional model \(\mathbf{b}\) of the bundle adjustment given in Equation (1) and the reference point determination model \(\mathbf{r}\) in Equation (18) are nonlinear equations. By solving a linear substitute problem during the least-squares adjustment, the estimates are generally biased, cf. Equations (24) and (25). To estimate a second-order correction, the spherical simplex unscented transformation is applied as described in Section 2.3. Based on the observed image coordinates and \(w_{0}=0\), \(n_{\text{obs}}+1\)\(\Sigma\)-points \(\mathcal{Y}\) are generated according to Equation (37). By concatenating both nonlinear functional models, i.e.,  
\[\mathcal{X}_{i}=\mathbf{r}(\mathbf{b}(\mathcal{Y}_{i})),\]  
these \(\Sigma\)-points are directly transformed. A second-order approximation of the reference point and the axis offset, and the related dispersion is obtained by Equations (29) and (30). Due to the large number of observations \(n_{\text{obs}}\) per measurement experiment, cf. Table 2, the SSUT is adopted once per configuration using the measurement experiments at DOY: 255, 258, 260.  
Table 4 shows the differences between the second-order approximation derived by the SSUT and the linear substitute solution derived by the GMM. The differences of the estimated parameters are close to zero. Moreover, the derived standard deviations are biased by about \(2\,\text{\mu m}\) and \(1\,\text{\mu m}\) for the coordinates of the reference point and the axis offset, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**DOY** & \(\delta\mathbf{X}\) & \(\delta\mathbf{Y}\) & \(\delta\mathbf{Z}\) & \(\mathbf{e_{\text{AO}}}\) & \(\hat{\mathbf{\sigma}}_{\mathbf{X}}\) & \(\hat{\mathbf{\sigma}}_{\mathbf{Y}}\) & \(\hat{\mathbf{\sigma}}_{\mathbf{Z}}\) & \(\hat{\mathbf{\sigma}}_{\mathbf{e_{\text{AO}}}}\) \\ \hline
255 & \(-0.01\) & \(0.02\) & \(0.03\) & \(-0.02\) & \(0.02\) & \(0.02\) & \(0.03\) & \(0.01\) \\
256 & \(0.03\) & \(-0.05\) & \(-0.08\) & \(-0.02\) & \(0.01\) & \(0.01\) & \(0.02\) & \(0.01\) \\
257 & \(0.01\) & \(0.05\) & \(-0.02\) & \(-0.01\) & \(0.02\) & \(0.02\) & \(0.03\) & \(0.01\) \\
258 & \(-0.02\) & \(-0.02\) & \(0.01\) & \(-0.02\) & \(0.02\) & \(0.02\) & \(0.03\) & \(0.01\) \\
259 & \(-0.06\) & \(0.13\) & \(0.07\) & \(-0.04\) & \(0.04\) & \(0.04\) & \(0.06\) & \(0.02\) \\
260 & \(0.01\) & \(-0.05\) & \(0.00\) & \(0.00\) & \(0.02\) & \(0.02\) & \(0.03\) & \(0.01\) \\
264 & \(0.06\) & \(-0.09\) & \(0.00\) & \(0.00\) & \(0.02\) & \(0.02\) & \(0.03\) & \(0.01\) \\ \hline \end{tabular}
\end{table}
Table 3: Variations of the estimated reference point position \(\mathbf{P}_{\text{IRP}}\), the derived axis offset \(e_{\text{AO}}\), and the related standard deviations. All estimates are given in mm.  
In comparison to the variations depicted in Figure 7, the bias can be neglected. The reason can be found in Equations (24) and (25). The second-order correction consists of the Hessian matrix, obtained by the functional model, and the dispersion matrix, defining the stochastic model. Doubtlessly, the nonlinearity comes from the functional model, but the stochastic model controls the size of the bias onto the estimates [30; 56]. Thanks to the high accuracy of close range photogrammetric systems but also due to the symmetric measurement configuration, the second-order effect becomes negligible, and the first-order solution yields proper estimates. This result confirms prior investigations using simulated data to evaluate reliable measurement configurations ([40] p. 86 ff).