# Sphero-Conical Modeling for the Estimation of Very Long Baseline Interferometry Invariant Point
## Appendix B Estimation of the IVP within the Gauss-Helmert Model  
Assuming the \"true\" coordinates for point \((x_{i},y_{i},z_{i})\) be \((\mu_{x},\mu_{y},\mu_{z})\), then the \(i\)-th pair of observed variables \((x_{i},y_{i},z_{i})\) (\(i=1\cdots n\)) is given by [16; 31; 18]  
\[ \begin{array}{llll}x_{i}&=\mu_{x_{i}}+e_{x_{i}},&E\left\{e_{x_{i}}\right\}&= 0\\ y_{i}&=\mu_{y_{i}}+e_{y_{i}},&E\left\{e_{y_{i}}\right\}&=0\\ z_{i}&=\mu_{z_{i}}+e_{z_{i}},&E\left\{e_{z_{i}}\right\}&=0.\end{array} \] (A3)  
If the random errors in the data are independent and identically distributed, the stochastic model of the random error vector can be represented as:  
\[ \begin{array}{ll}\boldsymbol{e}_{x}\\ \boldsymbol{e}_{y}\\ \boldsymbol{e}_{z}\end{array} \sim\left( \begin{array}{llll}\boldsymbol{0}\\ \boldsymbol{0}\\ \boldsymbol{0}\end{array} \right),\sigma_{0}^{2} \begin{bmatrix}I_{n}&0&0\\ 0&I_{n}&0\\ 0&0&I_{n}\end{bmatrix} =\sigma_{0}^{2}P^{-1}\right),\] (A4)  
where \(\sigma_{0}^{2}\) is the unitless variance component, \(P\) is a \(3n\times 3n\) weight matrix, and \(I_{n}\) is an \(n\times n\) identity matrix. In the case of the vertical motion given in Equation (A2), a (nonlinear) function that relates the \(i\)-th pair of variables to the unknown, nonrandom parameters \((\alpha,\beta,\gamma,c_{l})\) is provided by  
\[b_{i}(\alpha,\beta,\gamma,c_{l},\mu_{x_{i}},\mu_{y_{i}},\mu_{z_{i}})=(\mu_{x_{ i}}-\alpha)^{2}+(\mu_{y_{i}}-\beta)^{2}+(\mu_{z_{i}}-\gamma)^{2}-c_{l}^{2}=0.\] (A5)  
The function can be linearized about the approximate expansion point \((\alpha^{0},\beta^{0},\gamma^{0},\mu_{x_{i}}^{0},\mu_{y_{i}}^{0},\mu_{z_{i}}^ {0})\), resulting in partial derivatives for point \((x_{i},y_{i},z_{i})\) of the target \(k\) (azimuthal rotation) or target \(l\) (vertical rotation):  
\[\begin{array}{ll}b_{i}^{0}+2(\mu_{x_{i}}^{0}-\alpha^{0})d\mu_{x_{i}}+2(\mu_ {y_{i}}^{0}-\beta^{0})d\mu_{y_{i}}+2(\mu_{z_{i}}^{0}-\gamma^{0})d\mu_{z_{i}}\\ \phantom{=}-2(\mu_{x_{i}}^{0}-\alpha^{0})d\alpha-2(\mu_{y_{i}}^{0}-\beta^{0})d \beta-2(\mu_{z_{i}}^{0}-\gamma^{  
thus, upon substitution, the linearized equation can be rewritten as follows:  
\[ \begin{split} b_{i}^{0}&+2(\mu_{x_{i}}^{0}-\alpha^{0})( x_{i}-\mu_{x_{i}}^{0})+2(\mu_{y_{i}}^{0}-\beta^{0})(y_{i}-\mu_{y_{i}}^{0})+2(\mu_{z_{i}}^{0}- \gamma^{0})(z_{i}-\mu_{z_{i}}^{0})\\ &-2(\mu_{x_{i}}^{0}-\alpha^{0})e_{x_{i}}-2(\mu_{y_{i}}^{0}-\beta^ {0})e_{y_{i}}-2(\mu_{z_{i}}^{0}-\gamma^{0})e_{z_{i}}\\ &-2(\mu_{x_{i}}^{0}-\alpha^{0})d\alpha-2(\mu_{y_{i}}^{0}-\beta^{0 })d\beta-2(\mu_{z_{i}}^{0}-\gamma^{0})d\gamma-2c_{l}^{0}dc_{l}=0.\end{split}  \tag{10}\]  
Finally, we arrive at the equivalent form by rearranging  
\[ \begin{split}&\underbrace{b_{i}^{0}+2(\mu_{x_{i}}^{0}-\alpha^{0}) (x_{i}-\mu_{x_{i}}^{0})+2(\mu_{y_{i}}^{0}-\beta^{0})(y_{i}-\mu_{y_{i}}^{0})+2( \mu_{z_{i}}^{0}-\gamma^{0})(z_{i}-\mu_{z_{i}}^{0})}_{w_{i}}\\ &-\underbrace{\begin{bmatrix}2(\mu_{x_{i}}^{0}-\alpha^{0})&2(\mu_{y_{i}}^{0}- \beta^{0})&2(\mu_{z_{i}}^{0}-\gamma^{0})\\ \end{bmatrix}}_{B_{i}}\underbrace{\begin{bmatrix}e_{x_{i}}\\ e_{y_{i}}\\ e_{z_{i}}\end{bmatrix}}_{\mathbf{z_{i}}}\\ &-\underbrace{\begin{bmatrix}2(\mu_{x_{i}}^{0}-\alpha^{0})&2(\mu_{y_{i}}^{0}- \beta^{0})&2(\mu_{z_{i}}^{0}-\gamma^{0})&2r^{0}\\ \end{bmatrix}}_{A_{i}}\underbrace{\begin{bmatrix}d\alpha\\ d\beta\\ d\gamma\\ dc_{l}\end{bmatrix}}_{\mathbf{\xi}}=0,\end{split}  \tag{11}\]  
or, more compactly,  
\[w_{i}=A_{i}\mathbf{\xi}+B_{i}\mathbf{e}_{i}\quad(i=1\cdots n_{l}). \tag{12}\]  
Similarly, the mathematical model for the horizontal rotation can be represented by  
\[ \begin{split} a_{i}^{0}&+2(\mu_{x_{i}}^{0}-\alpha^{0})(x_{i}- \mu_{x_{i}}^{0})+2(\mu_{y_{i}}^{0}-\beta^{0})(y_{i}-\mu_{y_{i}}^{0})\\ &-\begin{bmatrix}2(\mu_{x_{i}}^{0}-\alpha^{0})&2(\mu_{y_{i}}^{0}- \beta^{0})&0\end{bmatrix}\begin{bmatrix}e_{x_{i}}\\ e_{y_{i}}\\ e_{z_{i}}\end{bmatrix}\\ &-\begin{bmatrix}2(\mu_{x_{i}}^{0}-\alpha^{0})&2(\mu_{y_{i}}^{0}- \beta^{0})&0&2r^{0}\end{bmatrix}\begin{bmatrix}d\alpha\\ d\beta\\ d\gamma\\ dc_{l}\end{bmatrix}=0.\end{split}