## 3 Evaluation of the CRLB with Calibration Targets
From the formulations of (7) and (13), the elements of the partial derivatives \(\partial\mathbf{r}^{\mathbf{o}}/\partial\mathbf{u}^{\mathbf{o}}\), \(\partial\mathbf{r}^{\mathbf{o}}/\partial\mathbf{s}^{\mathbf{o}}\), \(\partial\mathbf{r}_{\mathbf{c}}^{\mathbf{o}}/\partial\mathbf{c}^{\mathbf{o}}\) and \(\partial\mathbf{r}_{\mathbf{c}}^{\mathbf{o}}/\partial\mathbf{s}^{\mathbf{o}}\) in (17)-(21), can be determined as  
\[\frac{\partial\mathbf{r}^{\mathbf{o}}}{\partial\mathbf{u}^{\mathbf{o}}}(i_{m,n},1:3) =\frac{\left(\mathbf{u}^{\mathbf{o}}-\mathbf{s}_{\mathbf{u}}^{ \mathbf{o}}\right)^{\mathrm{T}}}{R_{\mathbf{t},m}^{\mathbf{o}}}+\frac{\left( \mathbf{u}^{\mathbf{o}}-\mathbf{s}_{\mathbf{r},n}^{\mathbf{o}}\right)^{ \mathrm{T}}}{R_{\mathbf{r},n}^{\mathbf{o}}} \tag{22}\]  
\[\frac{\partial\mathbf{r}^{\mathbf{o}}}{\partial\mathbf{s}^{\mathbf{o}}}(i_{m,n},3m-2:3m) =\frac{\left(\mathbf{s}_{\mathbf{t},m}^{\mathbf{o}}-\mathbf{u}^{ \mathbf{o}}\right)^{\mathrm{T}}}{R_{\mathbf{t},m}^{\mathbf{o}}}-\frac{\left( \mathbf{s}_{\mathbf{t},m}^{\mathbf{o}}-\mathbf{s}_{\mathbf{t},m}^{\mathbf{o} }\right)^{\mathrm{T}}}{R_{\mathbf{t},m,n}^{\mathbf{o}}} \tag{23}\]  
\[\frac{\partial\mathbf{r}_{\mathbf{c}}^{\mathbf{o}}}{\partial\mathbf{s}^{ \mathbf{o}}_{\mathbf{t}}}(i_{m,n},3m-2:3n) =\frac{\left(\mathbf{s}_{\mathbf{t},n}^{\mathbf{o}}-\mathbf{u}^{ \mathbf{o}}\right)^{\mathrm{T}}}{R_{\mathbf{t},n}^{\mathbf{o}}}-\frac{\left( \mathbf{s}_{\mathbf{r},n}^{\mathbf{o}}-\mathbf{s}_{\mathbf{t},m}^{\mathbf{o} }\right)^{\mathrm{T}}}{R_{\mathbf{t},m,n}^{\mathbf{o}}} \tag{24}\]  
\[\frac{\partial\mathbf{r}_{\mathbf{c}}^{\mathbf{o}}}{\partial\mathbf{s}^{ \mathbf{o}}_{\mathbf{t}}}(i_{m,n},3n-2:3n) =\frac{\left(\mathbf{s}_{\mathbf{t},n}^{\mathbf{o}}-\mathbf{u}^{ \mathbf{o}}\right)^{\mathrm{T}}}{R_{\mathbf{t},n}^{\mathbf{o}}}-\frac{\left( \mathbf{s}_{\mathbf{t},n}^{\mathbf{o}}-\mathbf{s}_{\mathbf{t},m}^{\mathbf{o} }\right)^{\mathrm{T}}}{R_{\mathbf{t},m,n}^{\mathbf{o}}} \tag{25}\]  
\[\frac{\partial\mathbf{r}_{\mathbf{c}}^{\mathbf{o}}}{\partial\mathbf{s}^{ \mathbf{o}}_{\mathbf{t}}}(i_{k,m,n},3m-2:3m) =\frac{\left(\mathbf{s}_{\mathbf{t},m}^{\mathbf{o}}-\mathbf{s}_{ \mathbf{t}}^{\mathbf{o}}\right)^{\mathrm{T}}}{R_{\mathbf{c},k,m}^{\mathbf{o}}}- \frac{\left(\mathbf{s}_{\mathbf{t},m}^{\mathbf{o}}-\mathbf{s}_{\mathbf{t},n}^{ \mathbf{o}}\right)^{\mathrm{T}}}{R_{\mathbf{t},m,n}^{\mathbf{o}}} \tag{26}\]  
\[\mathbf{Y}=\left(\frac{\partial\mathbf{r}^{\text{o}}}{\partial\mathbf{u}^{ \text{o}}}\right)^{\text{T}}\!\mathbf{Q}_{\mathbf{r}}^{-1}\!\left(\frac{ \partial\mathbf{r}^{\text{o}}}{\partial\mathbf{s}^{\text{o}}}\right) \tag{18}\]0  
\[\mathbf{Y}=\left(\frac{\partial\mathbf{r}^{\text{o}}}{\partial\mathbf{u}^{ \text{o}}}\right)^{\text{T}}\!\mathbf{Q}_{\mathbf{r}}^{-1}\!\left(\frac{ \partial\mathbf{r}^{\text{o}}}{\partial\mathbf{s}^{\text{o}}}\right) \tag{18}\]1  
\[\mathbf{Y}=\left(\frac{\partial\mathbf{r}^{\text{o}}}{\partial\mathbf{u}^{ \text{o}}}\right)^{\text{T}}\!\mathbf{Q}_{\mathbf{r}}^{-1}\!\left(\frac{ \partial\mathbf{r}^{\text{o}}}{\partial\mathbf{s}^{\text{o}}}\right) \tag{18}\]2  
for \(k=1,2,\ldots,K\), \(m=1,2,\ldots,M\) and \(n=1,2,\ldots,N\), and zeros elsewhere. By definition, the CRLB of \(\boldsymbol{\varphi}\), denoted by \(\mathbf{CRLB}_{\mathbf{c}}(\boldsymbol{\varphi})\), is given as \(\mathrm{FIM}(\boldsymbol{\varphi})^{-1}\), where only the upper left 3-by-3 block is for the target position \(\mathbf{u}^{\mathbf{o}}\).