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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

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

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

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

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

Due to the error term $\displaystyle{ {\boldsymbol \varepsilon} }$, in general $\displaystyle{ {\mathbf Y}={\mathbf G}\,{\mathbf X} }$ 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 $\displaystyle{ \hat{\mathbf X} }$ that minimises the discrepancy in the equations system. That is, the vector $\displaystyle{ \hat{\mathbf X} }$ providing the "best fit" of $\displaystyle{ {\mathbf Y} \simeq {\mathbf G}\,\hat{\mathbf X} }$ in a given sense.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $\displaystyle{ \hat{\mathbf X} }$ can be seen as an equilibrium solution.