If you wish to contribute or participate in the discussions about articles you are invited to join Navipedia as a registered user

Fundamentals
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Year of Publication 2011

The equation (1) neglects the measurement noise and missmodelling

$R^j=\rho^j+c(\delta t-\delta t^j)+T^j+\hat{\alpha}\, I^j+TGD^j+\mathcal{M}^j+{\boldsymbol \varepsilon}^j \qquad \mbox{(1)}$

If such errors (${\boldsymbol \varepsilon}$) are explicitly written, thence the lineal model is as follows:

${\mathbf Y}={\mathbf G}\,{\mathbf X}+{\boldsymbol \varepsilon} \qquad \mbox{(2)}$

where the error term ${\boldsymbol \varepsilon}$ is only known from some statistical properties, usually the mean ${\mathbf m}=E[{\boldsymbol \varepsilon}]$ and covariance matrix ${\mathbf R}=E[{\boldsymbol \varepsilon} \, {\boldsymbol \varepsilon}^T]$.

Due to the error term ${\boldsymbol \varepsilon}$, in general ${\mathbf Y}={\mathbf G}\,{\mathbf X}$ defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector $\hat{\mathbf X}$ that minimises the discrepancy in the equations system. That is, the vector $\hat{\mathbf X}$ providing the "best fit" of ${\mathbf Y} \simeq {\mathbf G}\,\hat{\mathbf X}$ in a given sense.

A common criterion used in GNSS is the Least-Squares adjustment, which is defined by the condition:

$\begin{array}{l} min \|{\mathbf Y}-\hat{\mathbf Y}\|^2=min \left[ \sum_{i=1}^n{\left ( y_i-\hat{y}_i \right )^2 }\right ] \qquad where \qquad \hat{\mathbf Y}={\mathbf G}\,\hat{\mathbf X} \qquad \mbox{(3)} \end{array}$

The discrepancy vector between the measurements ${\mathbf Y}$ and the fitted model $\hat{\mathbf Y }={\mathbf G}\,\hat{\mathbf X}$ is usually called the residual vector:

${\mathbf r}={\mathbf Y}-\hat{\mathbf Y}={\mathbf Y}-{\mathbf G}\,\hat{\mathbf X } \qquad \mbox{(4)}$

Thence, the Least-Squares estimator solution defined by equation (3), gives the vector $\hat{\mathbf X}$ that minimises [footnotes 1] the residuals quadratic norm $||{\mathbf r}||^2$.

From basic results of linear algebra, it follows that the solution fulfilling the condition (3) is given by:

$\hat{\mathbf X}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,{\mathbf Y} \qquad \mbox{(5)}$

Substituting (5) and (2) in (4) the post-fit -residual vector is:

${\mathbf r}=\left [\mathbf{I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\right]\, {\mathbf Y}= {\mathbf S}\, {\mathbf Y}={\mathbf S}\,{\boldsymbol \varepsilon}$

where ${\mathbf S}$ is a symmetrical, idempotent Projection matrix

${\mathbf S}={\mathbf I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,;\;\;{\mathbf S}^T={\mathbf S}\,\;\;;\;\;{\mathbf S}^2={\mathbf S}\,;\;\; {\mathbf r}= {\mathbf S}\, {\mathbf Y} \, {\boldsymbol \perp} \,\hat{\mathbf Y} \qquad \mbox{(6)}$

From (5) and (2) the estimator error can be written as:

${\mathbf{\Delta X}}=\hat{\mathbf X}-{\mathbf {X}}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,{\boldsymbol \varepsilon} \qquad \mbox{(7)}$

Assuming that the measurements minus model (i.e., prefit-residuals) have mean zero errors ($E[{\boldsymbol \varepsilon}]=0$) and covariance matrix ${\mathbf R}$, thence, the mean error, covariance matrix and Mean-Square Error (MSE) of the estimator are given by:

$\begin{array}{l} \begin{array}{rl} {\mathbf m}_{_{\mathbf \Delta X}} =&E[{\mathbf \Delta X}]=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,E[{\boldsymbol \varepsilon}]=0 \end{array}\\[0.3cm] \begin{array}{rl} {\mathbf P}_{_{\mathbf \Delta X}}= &E[{\mathbf \Delta X} \, {\mathbf \Delta X}^T]=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T \,E[{\boldsymbol \varepsilon}\, {\boldsymbol \varepsilon}^T]\,{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T \,\,{\mathbf R}\,\,{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1} \end{array} \\[0.3cm] \begin{array}{rl} \mbox{MSE}_{_{\mathbf \Delta X}}= &E[{\mathbf \Delta X}^T \, {\mathbf \Delta X}]= trace ({\mathbf P}_{_{\mathbf \Delta X}}) \end{array}~\\ \end{array} \qquad \mbox{(8)}$

The expression of ${\mathbf P}$ become simpler by assuming uncorrelated values with identical variance $\sigma^2$. That is, taking ${\mathbf R}=E[{\boldsymbol \varepsilon} \, {\boldsymbol \varepsilon}^T]=\sigma^2\,{\mathbf I}$, thence:

${\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(9)}$

1. ^ The equation (3), where a quadratic sum is minimised, could be interpreted in physical terms as minimising the energy of the error fit. Thence the estimate $\hat{\mathbf X}$ can be seen as an equilibrium solution.