## 4 Methodology
### The 9-Parameter Transformation
#### 4.5.1 Transformation Parameters  
For the linearization of the general expression, identical relations mentioned in 7-parameter and 8-parameter transformation (Equations (44)-(47)) can be applied. The difference between 7-parameter and 8-parameter transformation is an unequal scale factor for all axes. By introducing approximate values of the scale factor (\(\mu_{x}^{0}\), \(\mu_{y}^{0}\), \(\mu_{z}^{0}\)) and corresponding corrections (\(\delta\mu_{x}\), \(\delta\mu_{y}\), \(\delta\mu_{z}\)), the scale matrix \(\mathbf{S}\) for the 9-parameter transformation can be written as [8]  
\[\mathbf{S}=\mathbf{S}(\mu_{x},\mu_{y},\mu_{z})= \begin{bmatrix}\mu_{x}^{0}+ \delta\mu_{x}&0&0\\ 0&\mu_{y}^{0}+\delta\mu_{y}&0\\ 0&0&\mu_{z}^{0}+\delta\mu_{z}\end{bmatrix} =\mathbf{S}^{0}+\mathbf{M}\delta \mu_{x}+\mathbf{N}\delta\mu_{y}+\mathbf{P}\delta\mu_{z} \tag{66}\]  
where  
\[\mathbf{S}^{0}= \begin{bmatrix}\mu_{x}^{0}&0&0\\ 0&\mu_{y}^{0}&0\\ 0&0&\mu_{z}^{0}\end{bmatrix} \mathbf{M}= \begin{bmatrix}1&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix} \mathbf{N}= \begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix} \mathbf{P}= \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&1\end{bmatrix} \]  
Introducing Equations (47) and (66) in the general expression for the 9-parameter transformation (Equation (64)) yields  
\[\mathbf{X}_{o}=\mathbf{t}+(\mathbf{R}^{0}+\mathbf{E}\delta\alpha_{x}+\mathbf{ F}\delta\alpha_{y}+\mathbf{G}\delta\alpha_{z})\cdot(\mathbf{S}^{0}+\mathbf{M} \delta\mu_{x}+\mathbf{N}\delta\mu_{y}+\mathbf{P}\delta\mu_{z})\cdot\mathbf{X }_{i} \tag{67}\]  
Figure 7: The 9-parameter transformation. As \(\delta\alpha_{i}\) and \(\delta\mu_{i}\) have very small values, their product will be negligible, so the observation equations can be written as [8]  
\[\mathbf{v}=\mathbf{t}+(\mathbf{R}^{0}\mathbf{S}^{0}+\mathbf{R}^{0}\mathbf{M} \delta\mu_{x}+\mathbf{R}^{0}\mathbf{N}\delta\mu_{y}+\mathbf{R}^{0}\mathbf{P} \delta\mu_{z}+\mathbf{ES}^{0}\delta\alpha_{x}+\mathbf{FS}^{0}\delta\alpha_{y}+ \mathbf{GS}^{0}\delta\alpha_{z})\cdot\mathbf{X}_{i}-\mathbf{X}_{o} \tag{68}\]  
Based on the observation equations, design matrix \(\mathbf{A}\) and observation vector \(\mathbf{I}\) can be formed:  
\[\mathbf{A}= \begin{bmatrix}1&0&0&a_{14}^{1}&a_{15}^{1}&a_{16}^{1}&a_{17}^{1}&a_ {18}^{1}&a_{19}^{1}\\ 0&1&0&a_{34}^{1}&a_{35}^{1}&a_{36}^{1}&a_{37}^{1}&a_{38}^{1}&a_{39}^{1}\\ 0&0&1&a_{34}^{1}&a_{35}^{1}&a_{36}^{1}&a_{37}^{1}&a_{38}^{1}&a_{39}^{1}\\ &&\vdots&&\vdots&&\\ 1&0&0&a_{14}^{1}&a_{35}^{1}&a_{36}^{1}&a_{37}^{1}&a_{38}^{1}&a_{39}^{1}\\ 0&1&0&a_{34}^{1}&a_{35}^{1}&a_{35}^{1}&a_{36}^{1}&a_{37}^{1}&a_{38}^{1}&a_{39} ^{1}\\ 0&1&0&a_{34}^{1}&a_{35}^{1}&a_{35}^{1}&a_{36}^{1}&a_{37}^{1}&a_{38}^{1}&a_{39} ^{1}\\ \end{bmatrix}  \tag{69}\]  
It is also important to know what the order of unknown parameters is ingen matrix \(\mathbf{A}\) is. In this case, the order of the unknown parameters is \(t_{x}\), \(t_{y}\), \(t_{z}\), \(\alpha_{x}\), \(\alpha_{y}\), \(\alpha_{z}\), \(\mu_{x}\), \(\mu_{y}\), and \(\mu_{z}\), so coefficients \(a_{ij}\) in design matrix \(\mathbf{A}\) have the following values:  
\[ \begin{array}{ll}a_{14}^{i}=-\mu_{y}^{0}r_{13}^{0}Y_{i}^{i}+\mu_{z}^{0}r_{12 }^{0}Z_{i}^{i}&a_{17}^{i}=r_{11}^{0}X_{i}^{i}\\ a_{24}^{i}=-\mu_{y}^{0}r_{23}^{0}Y_{i}^{i}+\mu_{z}^{0}r_{22}^{0}Z_{i}^{i}&a_{2 7}^{i}=r_{21}^{0}X_{i}^{i}\\ a_{34}^{i}=-\mu_{y}^{0}r_{33}^{0}Y_{i}^{i}+\mu_{z}^{0}r_{32}^{0}Z_{i}^{i}&a_{3 7}^{i}=r_{31}^{0}X_{i}^{i}\\ a_{15}^{i}=-\cos a_{z}^{0}(\mu_{x}^{0}\gamma_{31}^{0}X_{i}^{i}+\mu_{y}^{0}r_{ 32}^{0}Y_{i}^{i}+\mu_{z}^{0}r_{33}^{0}Z_{i}^{i})&a_{18}^{i}=r_{12}^{0}Y_{i}^{i }\\ a_{25}^{i}=\sin a_{z}^{0}(\mu_{x}^{0}\gamma_{31}^{0}X_{i}^{i}+\mu_{y}^{0}r_{ 32}^{0}Y_{i}^{i}+\mu_{z}^{0}r_{33}^{0}Z_{i}^{i})&a_{28}^{i}=r_{22}^{0}Y_{i}^{i }\\ a_{35}^{i}=\mu_{x}^{0}\cos a_{y}^{0}X_{i}^{i}+\mu_{y}^{0}r_{31}^{0}\sin a_{x}^ {0}Y_{i}^{i}-\mu_{x}^{0}r_{31}^{0}\cos a_{x}^{0}Z_{i}^{i}&a_{38}^{i}=r_{32}^{0 }Y_{i}^{i}\\ \end{array} \]  
\[