## Appendix A  
The first partial derivative of Equation (8) is  
\[\frac{\partial\ln L}{\partial\mathbf{x}}=-\frac{1}{2\sigma^{2}}\frac{\partial }{\partial\mathbf{x}}\Big{(}(\widetilde{\mathbf{r}}-\mathbf{d}(\mathbf{x}))^ {T}(\widetilde{\mathbf{r}}-\mathbf{d}(\mathbf{x}))\Big{)}=\frac{1}{\sigma^{2} }\frac{\partial\mathbf{r_{g}}(\mathbf{x})^{T}}{\partial\mathbf{x}}\mathbf{H} ^{T}(\widetilde{\mathbf{r}}-\mathbf{d}(\mathbf{x})) \tag{10}\]  
where  
\[\mathbf{r}_{g}(\mathbf{x})=\big{[}\|\mathbf{x}_{r}-\mathbf{x}\|\quad\| \mathbf{x}_{1}^{t}-\mathbf{x}\|\quad\quad\ldots\quad\big{\|}\mathbf{x}_{N}^{t} -\mathbf{x}\|\big{]}^{T}, \tag{11}\]  
and  
\[\mathbf{H}=\big{[}\mathbf{1}_{N}\mathbf{I}_{N}\big{]}=\begin{bmatrix}1&1& \cdots&0&0\\ 1&0&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 1&0&\cdots&0&1\end{bmatrix}. \tag{12}\]  
Based on the expression in Equation (10), we conclude that the componental partial derivative is  
\[\frac{\partial\ln L}{\partial x}=\frac{1}{\sigma^{2}}\bigg{(}\frac{\partial \mathbf{r}_{g}(\mathbf{x})}{\partial x}\bigg{)}^{T}\mathbf{H}^{T}(\widetilde {\mathbf{r}}-\mathbf{d}(\mathbf{x})). \tag{13}\]  
Then, the first element of matrix \(\mathrm{F}(\mathbf{x})\) is  
\[\begin{split} F_{11}&=E\Bigg{[}\frac{\partial\ln L}{ \partial x}\bigg{(}\frac{\partial\ln L}{\partial x}\bigg{)}^{T}\Bigg{]}\\ &=E\Bigg{[}\frac{1}{\sigma^{4}}\bigg{(}\frac{\partial\mathbf{r }_{g}(\mathbf{x})}{\partial x}\bigg{)}^{T}\mathbf{H}(\widetilde{\mathbf{r}}- \mathbf{d}(\mathbf{x}))(\widetilde{\mathbf{r}}-\mathbf{d}(\mathbf{x}))^{T} \mathbf{H}\bigg{(}\frac{\partial\mathbf{r}_{g}(\mathbf{x})}{\partial x}\bigg{)} \Bigg{]}\\ &=\frac{1}{\sigma^{2}}\bigg{(}\frac{\partial\mathbf{r}_{g}( \mathbf{x})}{\partial x}\bigg{)}^{T}\mathbf{H}^{T}\mathbf{H}\bigg{(}\frac{ \partial\mathbf{r}_{g}(\mathbf{x})}{\partial x}\bigg{)}.\end{split} \tag{14}\]  
Analogus to this the remaining elements of \(\mathrm{F}(\mathbf{x})\) are  
\[F_{12}=F_{21}=E\Bigg{[}\bigg{(}\frac{\partial\ln L}{\partial x}\bigg{)}\bigg{(} \frac{\partial\ln L}{\partial y}\bigg{)}^{T}\Bigg{]}=\frac{1}{\sigma^{2}} \bigg{(}\frac{\partial\mathbf{r}_{g}(\mathbf{x})}{\partial x}\bigg{)}^{T} \mathbf{H}^{T}\mathbf{H}\bigg{(}\frac{\partial\mathbf{r}_{g}(\mathbf{x})}{ \partial y}\bigg{)}, \tag{15}\]  
\[F_{22}=E\Bigg{[}\bigg{(}\frac{\partial\ln L}{\partial y}\bigg{)}\bigg{(}\frac{ \partial\ln L}{\partial y}\bigg{)}^{T}\Bigg{]}=\frac{1}{\sigma^{2}}\bigg{(} \frac{\partial\mathbf{r}_{g}(\mathbf{x})}{\partial y}\bigg{)}^{T}\mathbf{H}^{T} \mathbf{H}\bigg{(}\frac{\partial\mathbf{r}_{g}(\mathbf{x})}{\partial y}\bigg{)}. \tag{16}\]If we take \(\frac{\partial\mathbf{r}_{g}(\mathbf{x})}{\partial x}\), the following is obtained  
\[\frac{\partial\mathbf{r}_{g}(\mathbf{x})}{\partial x}=\left[\begin{array}{c} \frac{x-x_{r}}{\|\mathbf{x}-\mathbf{x}_{r}\|_{2}}&\frac{y-y_{r}}{\|\mathbf{x}- \mathbf{x}_{r}\|_{2}}\\ \frac{x-x_{r}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{r}^{\prime}\|_{2}}&\frac{y-y _{r}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{r}^{\prime}\|_{2}}\\ \vdots&\vdots\\ \frac{x-x_{r}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{r}^{\prime}\|_{2}}&\frac{y- y_{r}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{r}^{\prime}\|_{2}}\end{array}\right], \tag{100}\]  
Then, the expressions in Equations (101)-(102) are simplified to  
\[F_{11}=\frac{1}{\sigma^{2}}\sum_{i=1}^{N}\left(\frac{x-x_{r}}{\|\mathbf{x}- \mathbf{x}_{r}\|_{2}}+\frac{x-x_{i}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{i}^{ \prime}\|_{2}}\right)^{2}, \tag{101}\]  
\[F_{12}=F_{21}=\frac{1}{\sigma^{2}}\sum_{i=1}^{N}\left(\frac{x-x_{r}}{\| \mathbf{x}-\mathbf{x}_{r}\|_{2}}+\frac{x-x_{i}^{\prime}}{\|\mathbf{x}- \mathbf{x}_{i}^{\prime}\|_{2}}\right)\left(\frac{y-y_{r}}{\|\mathbf{x}- \mathbf{x}_{r}\|_{2}}+\frac{y-y_{i}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{i}^{ \prime}\|_{2}}\right), \tag{102}\]  
\[F_{22}=\frac{1}{\sigma^{2}}\sum_{i=1}^{N}\left(\frac{y-y_{r}}{\|\mathbf{x}- \mathbf{x}_{r}\|_{2}}+\frac{y-y_{i}^{\prime}}{\|\mathbf{x}-\mathbf{x}_{i}^{ \prime}\|_{2}}\right)^{2}. \tag{103}\]