## 3 Theoretical Basis of the Method of Least Squares with Conditional Equations
Using \(\mathbf{v}=[v_{1},v_{2}\cdots,v_{n}]^{T}\), the vector of random observation errors with the covariance matrix \(\mathbf{C}_{\mathbf{v}}=\sigma_{0}^{2}\mathbf{Q}=\sigma_{0}^{2}\mathbf{P}^{-1}\) and the vector of expected values \(E(\mathbf{v})=0\) (\(\mathbf{Q}\)--known cofactor matrix, \(\mathbf{P}\)--weight matrix, \(\sigma_{0}^{2}\)--unknown variance coefficient). When we assume that the approximate coordinates \(\mathbf{X}^{0}=\left[X_{1}^{0},\ldots,X_{r}\right]^{T}\) of GNSS receiver antennas are known, then the equation \(\mathbf{y}+\mathbf{v}\)=\(\mathbf{F}(\mathbf{X})\) can be replaced with a linear observation equation with the following form:  
\[\mathbf{v}=\mathbf{F}(\mathbf{X})-\mathbf{y}=\mathbf{F}\Big{(}\mathbf{X}^{0} +d\mathbf{X}\Big{)}-\mathbf{y}=\left.\frac{\partial\mathbf{F}(\mathbf{X})}{ \partial\mathbf{X}}\right|_{\mathbf{X}=\mathbf{X}^{0}}d\mathbf{X}+\mathbf{F} \Big{(}\mathbf{X}^{0}\Big{)}-\mathbf{y}=\mathbf{A}d\mathbf{X}+\mathbf{1} \tag{3}\]  
The quantity \(d\mathbf{X}\) is the vector of unknown coordinate increments, such that \(\mathbf{X}=\mathbf{X}^{0}+d\mathbf{X}\). The matrix  
\[\mathbf{A}=\left.\frac{\partial\mathbf{F}(\mathbf{X})}{\partial\mathbf{X}} \right|_{\mathbf{X}=\mathbf{X}^{0}}=\left[ \begin{array}{cccc}\frac{\partial \bar{r}_{1}(\mathbf{X})}{\partial X_{1}}&\frac{\partial\bar{r}_{1}(\mathbf{X} )}{\partial X_{2}}&\ldots&\frac{\partial\bar{r}_{1}(\mathbf{X})}{\partial X _{r}}\\ \frac{\partial\bar{r}_{2}(\mathbf{X})}{\partial X_{1}}&\frac{\partial\bar{r}_{2 }(\mathbf{X})}{\partial X_{2}}&\ldots&\frac{\partial\bar{r}_{2}(\mathbf{X})}{ \partial X_{r}}\\ \ldots&\ldots&\ldots&\ldots\\ \frac{\partial\bar{r}_{n}(\mathbf{X})}{\partial X_{1}}&\frac{\partial\bar{r}_{n }(\mathbf{X})}{\partial X_{2}}&\ldots&\frac{\partial\bar{r}_{n}(\mathbf{X})}{ \partial X_{r}}\end{array} \right]_{\mathbf{X}=\mathbf{X}^{0}}\]  
is the design of the system of linear equations, while  
\[\mathbf{1}=\mathbf{F}\Big{(}\mathbf{X}^{0}\Big{)}-\mathbf{y}=\left[F_{1} \Big{(}\mathbf{X}^{0}\Big{)}-y_{1},\;F_{2}\Big{(}\mathbf{X}^{0}\Big{)}-y_{2}, \;\cdots,\;F_{n}\Big{(}\mathbf{X}^{0}\Big{)}-y_{n}\right]^{T}\]  
is the absolute term vector. Let us also assume that the matrix \(\mathbf{A}\) is a column full rank, i.e., \(rank(\mathbf{A})=r\). Considering that \(\mathbf{X}=\mathbf{X}^{0}+d\mathbf{X}\), the system of observation equations (1) can also be similarly brought to the linear form. After developing the function \(\mathbf{\Psi}(\mathbf{X})\) to the form  
\[\Psi(\mathbf{X})=\Psi\Big{(}\mathbf{X}^{0}+d\mathbf{X}\Big{)}=\Psi\Big{(} \mathbf{X}^{0}\Big{)}+\left.\frac{\partial\Psi(\mathbf{X})}{\partial\mathbf{X }}\right|_{\mathbf{X}=\mathbf{X}^{0}}d\mathbf{X}=\Psi\Big{(}\mathbf{X}^{0} \Big{)}+\mathbf{B}d\mathbf{X} \tag{4}\]  
the following linear conditional equation is obtained:  
\[\mathbf{B}d\mathbf{X}+\Delta=0 \tag{5}\]where:  
\[\left.\begin{array}{l}\mathbf{B}=\left.\frac{\partial\Psi(\mathbf{X})}{\partial \mathbf{X}}\right|_{\mathbf{X}=\mathbf{X}^{0}}=\left[ \begin{array}{cccc} \frac{\partial\mathbf{\tau}_{1}(\mathbf{X})}{\partial\mathbf{X}}\frac{ \partial\mathbf{\tau}_{1}(\mathbf{X})}{\partial\mathbf{X}}\dots&\frac{ \partial\mathbf{\tau}_{1}(\mathbf{X})}{\partial\mathbf{X}}\\ \frac{\partial\mathbf{\tau}_{2}(\mathbf{X})}{\partial\mathbf{X}_{1}}\frac{ \partial\mathbf{\tau}_{2}(\mathbf{X})}{\partial\mathbf{X}_{2}}\dots&\frac{ \partial\mathbf{\tau}_{2}(\mathbf{X})}{\partial\mathbf{X}_{x}}\\ \dots&\dots&\dots&\dots\\ \frac{\partial\mathbf{\tau}_{w}(\mathbf{X})}{\partial\mathbf{X}_{1}}\frac{ \partial\mathbf{\tau}_{w}(\mathbf{X})}{\partial\mathbf{X}_{2}}\dots&\frac{ \partial\mathbf{\tau}_{w}(\mathbf{X})}{\partial\mathbf{X}_{x}}\end{array}