# Use of Total Least Squares Adjustment in Geodetic Applications
## 2 Solution of the Predetermined Model  
### Motivation  
Measurements and unknowns are associated with mathematical expressions in the model of observations equations, Equation (1), where we connect them in linearized form [2]:  
\[\mathbf{A}\mathbf{x}=\mathbf{b}, \begin{bmatrix}a_{11}&\cdots&a_{1u}\\ \vdots&\ddots&\vdots\\ a_{n1}&\cdots&a_{nu}\end{bmatrix} \cdot \begin{bmatrix}x_{1}\\ \vdots\\ x_{u}\end{bmatrix} = \begin{bmatrix}b_{1}\\ \vdots\\ b_{n}\end{bmatrix} , \tag{1}\]  
where \(n\) is the number of measurements or the number of all equations of the considered model, \(u\) is the number of unknowns, \(\mathbf{b}\) is the vector of observations or measurements, \(\mathbf{A}\) is the design or the model matrix, which consists of coefficients belonging to each unknown and is treated as a forward operator, and \(\mathbf{x}\) is the vector of unknowns. In model (1), the vector of observations \(\mathbf{b}\) lies in multidimensional space, spanned by the columns of matrix \(\mathbf{A}\)[2]. Rank of matrix \(\mathbf{A}\) is \(\mathrm{rank}(\mathbf{A})=r\leq\min(n,u)\). In system (1), connecting unknowns and measurements, there are three possible situations for a system of \(n\) equations for \(u\) unknowns: a predetermined system when \(n>u\), a uniquely determined system when \(n=u\), and an underdetermined system when \(n<u\) (the latter is not common when measurements are involved, since we always aim for a redundant number of measurements). System (1) can also be written in the form:  
\[[\mathbf{A}|\mathbf{b}] \begin{bmatrix}\mathbf{x}\\ -\mathbf{1}\end{bmatrix} =\mathbf{0}. \tag{2}\]Let us focus on a predetermined system and present it through finding the intersection of three lines in Figure 1:  
* Three lines intersect at the same point T--the left in Figure 1: In this case, we deal with the system of \(n\) consistent equations. The property of system (1) is that \(r=\text{rank}(\mathbf{A})=\text{rank}([\mathbf{A}|\mathbf{b}])=u\). In geometrical representation, vector \(\mathbf{b}\) lies in a space spanned by the columns of matrix \(\mathbf{A}\); or in other words, vector \(\mathbf{b}\) can be represented as a linear combination of the columns of matrix \(\mathbf{A}\). * Three lines (blue) do not intersect at the same point--the right in Figure 1: In this case, we have \(r=\text{rank}(\mathbf{A})<\text{rank}([\mathbf{A}|\mathbf{b}])=u+1\) and the equality in Equation (1) does not apply. We can write \(\mathbf{A}\mathbf{x}\cong\mathbf{b}\) instead. In geometrical representation, the observation vector \(\mathbf{b}\) does not lie in the space spanned by the column vectors of matrix \(\mathbf{A}\) or cannot be written as a linear combination of the column vectors of matrix \(\mathbf{A}\). There is no unique solution as we have three non-consistent equations. The aim is to find the orthogonal projection of vector \(\mathbf{b}\) in the above mentioned space; or in other words, our aim is to transform system (1) to the consistent form with the solution for \(\mathbf{x}\rightarrow\hat{\mathbf{x}}\) and consequently \(\mathbf{b}\rightarrow\hat{\mathbf{b}}\) (the right in Figure 1, red lines), where \(\mathbf{A}\hat{\mathbf{x}}=\hat{\mathbf{b}}\), for which it would apply that \(r=\text{rank}(\mathbf{A})=\text{rank}([\mathbf{A}\hat{\mathbf{b}}])=u\). Therefore, we want to reduce the system rank of \([\mathbf{A}|\mathbf{b}]\) by 1. In addition, for the predetermined system of non-consistent equations, we search for a solution, which will fulfill the requirement of the minimal sum of the squares of measurement residuals \(\left\|\mathbf{v}\right\|^{2}=\mathbf{v}^{\text{T}}\mathbf{P}\mathbf{v}=\text{ min. }\), where \(\mathbf{v}=\hat{\mathbf{b}}-\mathbf{b}\) are measurement residuals and \(\mathbf{P}\) is the weight matrix of measurements.