## 4 Methodology
### The 7-Parameter Transformation
#### 4.3.1 Transformation Parameters  
From the expressions for rotation matrix, it is clear that Equation (29) is not a linear function, and it is necessary to linearize it to form design matrix \(\mathbf{A}\) and observation vector \(\mathbf{1}\)[8]. Due to complexity of linearization, it will be shown only for the case where the output coordinate system rotates around the input one (CFR convention); for the second case, similar procedure applies. To linearize Equation (29), approximate values of rotation angles (\(\alpha_{x}^{0}\), \(\alpha_{y}^{0}\), \(\alpha_{z}^{0}\)) and corresponding corrections (\(\delta\alpha_{x}\), \(\delta\alpha_{y}\), \(\delta\alpha_{z}\)) must be introduced. Definite values of rotation angles can be obtained by [8]:  
\[\alpha_{x} =\alpha_{x}^{0}+\delta\alpha_{x}\] \[\alpha_{y} =\alpha_{y}^{0}+\delta\alpha_{y} \tag{44}\] \[\alpha_{z} =\alpha_{z}^{0}+\delta\alpha_{z}\]  
Based on approximate values of rotation angles, an approximate rotation matrix can be calculated. In abridged form, it can be expressed as [8]  
\[\mathbf{R}^{0}= \begin{bmatrix}r_{11}^{0}&r_{12}^{0}&r_{13}^{0}\\ r_{11}^{0}&r_{22}^{0}&r_{33}^{0}\end{bmatrix}  \tag{45}\]  
Elements of the rotation matrix \(\mathbf{R}\) are functions of rotation angles that can be linearized by approximated values of rotation angles [8]:  
\[r_{ij}(\alpha_{x},\alpha_{y},\alpha_{z})=r_{ij}(\alpha_{x}^{0},\alpha_{y}^{0},\alpha_{z}^{0})+\frac{\partial r_{ij}}{\partial\alpha_{x}}\delta\alpha_{x}+ \frac{\partial r_{ij}}{\partial\alpha_{y}}\delta\alpha_{y}+\frac{\partial r_{ ij}}{\partial\alpha_{z}}\delta\alpha_{z} \tag{46}\]  
where all derivatives from Equation (46) must be evaluated for approximate rotation angles. In matrix form, Equation (46) is as follows [8]:  
\[\mathbf{R}=\mathbf{R}^{0}+\mathbf{E}\delta\alpha_{x}+\mathbf{F}\delta\alpha_{ y}+\mathbf{G}\delta\alpha_{z} \tag{47}\]  
where matrices \(\mathbf{E}\), \(\mathbf{F}\), and \(\mathbf{G}\) are shown in Appendix A.3. Scale factor can also be decomposed into two parts--approximate value of scale factor (\(\mu^{0}\)) and corresponding correction (\(\delta\mu\))--so a definite value of scale factor can be calculated as [8]  
\[\mu=\mu^{0}+\delta\mu \tag{48}\]  
Including terms from Equations (47) and (48) into Equation (29) yields [8]  
\[\mathbf{X}_{o}=\mathbf{t}+(\mathbf{R}^{0}+\mathbf{E}\delta\alpha_{x}+\mathbf{ F}\delta\alpha_{y}+\mathbf{G}\delta\alpha_{z})\cdot(\mu^{0}+\delta\mu)\cdot \mathbf{X}_{i} \tag{49}\]  
Because \(\delta\alpha_{i}\) and \(\delta\mu\) have very small values, their mutual multiplication is negligible, so the observation equation in matrix form can be obtained [8]:  
\[\mathbf{v}=\mathbf{t}+\mathbf{R}^{0}\delta\mu\mathbf{X}_{i}+\mathbf{E}\mu^{0 }\mathbf{X}_{i}\delta\alpha_{x}+\mathbf{F}\mu^{0}\mathbf{X}_{i}\delta\alpha_{y }+\mathbf{G}\mu^{0}\mathbf{X}_{i}\delta\alpha_{z}+\mathbf{R}^{0}\mu^{0} \mathbf{X}_{i}-\mathbf{X}_{o} \tag{50}\]  
Based on the calculated inverses of the rotation matrices, it is evident that the rotation matrix is orthonormal, which means that \(R^{-1}=R^{T}\). To illustrate the purpose of a reversible transformation, an example of transforming a point from HDKS to HTRS96 and back to HDKS based on two sets of parameters and the reversible transformation is shown. Based on 5000 identical points in the HDKS and HTRS96, it is possible to calculate two sets of parameters:  
* From HDKS to HTRS96: \begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(t_{x}\) (m) & 546.509 \\ \(t_{y}\) (m) & 162.269 \\ \(t_{z}\) (m) & 469.395 \\ \(\mu\) (ppm) & \(-\)4.417 \\ \(r_{x}\) (\({}^{\prime\prime}\)) & \(-\)5.906 \\ \(r_{y}\) (\({}^{\prime\prime}\)) & \(-\)2.075 \\ \(r_{z}\) (\({}^{\prime\prime}\)) & 11.507 \\ \hline \hline \end{tabular} 
* From HTRS96 to HDKS: