## 4 Methodology
### 8-Parameter Transformation
#### 4.4.1 Transformation Parameters  
Content of the translation vector and rotation matrix were defined in the previous section. The difference compared to the 7-parameter transformation is the scale factor that, in this case, cannot be written as a scalar, but as a matrix because scale factor is not unique for all axes. By introducing approximate values of the scale factor and corresponding corrections, the scale matrix \(\mathbf{S}\) can be written as [8]  
\[\mathbf{S}=\mathbf{S}(\mu_{xy},\mu_{z})= \begin{bmatrix}\mu_{xy}^{0}+\delta\mu_ {xy}&0&0\\ 0&\mu_{xy}^{0}+\delta\mu_{xy}&0\\ 0&0&\mu_{z}^{0}+\delta\mu_{z}\end{bmatrix} =\mathbf{S}^{0}+\mathbf{M}\delta\mu _{xy}+\mathbf{P}\delta\mu_{z} \tag{58}\]  
where \(\mathbf{M}\) and \(\mathbf{P}\) are  
\[\mathbf{M}= \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix} \quad\mathbf{P}= \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&1\end{bmatrix} \]  
Rotation matrix \(\mathbf{R}\) was defined in Equation (47). Introducing Equations (47) and (58) into (56) yields  
\[\mathbf{X}_{o}=\mathbf{t}+(\mathbf{R}^{0}+\mathbf{E}\delta\alpha_{x}+\mathbf{ F}\delta\alpha_{y}+\mathbf{G}\delta\alpha_{z})\cdot(\mathbf{S}^{0}+\mathbf{M} \delta\mu_{xy}+\mathbf{P}\delta\mu_{z})\cdot\mathbf{X}_{i} \tag{59}\]  
The product of elements \(\delta\alpha_{i}\) and \(\delta\mu_{i}\) has negligible value, so Equation (59) can be simplified. From the simplified expression, observation equations can be formed and written [8]:  
\[\mathbf{v}=\mathbf{t}+\mathbf{R}^{0}\mathbf{M}\mathbf{X}_{i}\delta\mu_{xy}+ \mathbf{R}^{0}\mathbf{P}\mathbf{X}_{i}\delta\mu_{z}+\mathbf{E}\mathbf{S}^{0} \mathbf{X}_{i}\delta\alpha_{x}+\mathbf{F}\mathbf{S}^{0}\mathbf{X}_{i}\delta \alpha_{y}+\mathbf{G}\mathbf{S}^{0}\mathbf{X}_{i}\delta\alpha_{z}+\mathbf{R}^ {0}\mathbf{S}^{0}\mathbf{X}_{i}-\mathbf{X}_{o} \tag{60}\]  
Based on observation equations, design matrix \(\mathbf{A}\) and vector \(\mathbf{I}\) can be formed:  
\[\mathbf{A}= \begin{bmatrix}1&0&0&a_{14}^{1}&a_{15}^{1}&a_{16}^{1}&a_{17}^{1}&a _{18}^{1}\\ 0&1&0&a_{24}^{1}&a_{25}^{1}&a_{26}^{1}&a_{27}^{1}&a_{28}^{1}\\ 0&0&1&a_{34}^{1}&a_{35}^{1}&a_{36}^{1}&a_{37}^{1}&a_{38}^{1}\\ &&\vdots\\ 1&0&0&a_{14}^{n}&a_{15}^{n}&a_{16}^{n}&a_{17}^{n}&a_{18}^{n}\\ 0&1&0&a_{24}^{n}&a_{25}^{n}&a_{26}^{n}&a_{27}^{n}&a_{28}^{n}\\ 0&0&1&a_{34}^{n}&a_{35}^{n}&a_{36}^{n}&a_{37}^{n}&a_{38}^{n}\\ \end{bmatrix} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad 1= \begin{bmatrix}X_{1}^{1}-\mu_{xy} \mu_{11}^{0}X_{1}^{1}-\mu_{xy}\mu_{12}^{0}Y_{1}^{1}-\mu_{x}\mu_{13}^{0}Z_{1} ^{1}\\ Y_{1}^{1}-\mu_{xy}\mu_{21}^{0}X_{1}^{1}-\mu_{xy}\mu_{22}^{0}Y_{1}^{1}-\mu_{x} \mu_{23}^{0}Z_{1}^{1}\\ Z_{0}^{1}-\mu_{xy}\mu_{31}^{0}X_{1}^{1}-\mu_{xy}\mu_{32}^{0}Y_{1}^{1}-\mu_{x} \mu_{33}^{0}Z_{1}^{1}\\ &\vdots\\ X_{o}^{n}-\mu_{xy}\mu_{11}^{0}X_{1}^{n}-\mu_{xy}\mu_{12}^{0}Y_{1}^{n}-\mu_{x} \mu_{13}^{0}Z_{1}^{n}\\ Y_{o}^{n}-\mu_{xy}\mu_{21}^{0}X_{1}^{n}-\mu_{xy}\mu_{22}^{0}Y_{1}^{n}-\mu_{x} \mu_{23}^{0}Z_{1}^{n}\\ Z_{o}^{n}-\mu_{xy}\mu_{31}^{0}X_{1}^{n}-\mu_{xy}\mu_{32}^{0}Y_{1}^{n}-\mu_{x} \mu_{33}^{0}Z_{1}^{n}\end{bmatrix}  \tag{61}\]  
It is also important to know what order of unknown parameters in the design matrix [8].