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Weighted Least Square Solution (WLS)

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

In the least squares fit

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

all measurements have been treated in the same way (see Parameters adjustment). Nevertheless in a real scenario, not all the measurements have the same quality, i.e., the same error. For instance, measurements at low elevation have larger multipath error than the measurements at high elevation. In the same way, the modelling errors due to the tropospheric or ionospheric mapping functions are larger at lower elevations.

The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix ${\mathbf W}$ and re-defining the "best-fit" as the condition [footnotes 1]:

$min \|{\mathbf Y}-{\mathbf \hat{Y}}\|^{^{ 2}}_{_{\mathbf W}} \qquad \mbox{(2)}$

where the norm of the residuals vector ${\mathbf r}={\mathbf Y}-{\mathbf \hat{Y}}$ to minimise is, now, associated to the scalar product defined by the weighting matrix ${\mathbf W}$.

With this weighting criteria, estimator $\hat{\mathbf X}_{_{\mathbf W}}$ and its covariance matrix ${\mathbf P}_{_{\mathbf W}}$ are:

$\hat{\mathbf X}_{_{\mathbf W}}=({\mathbf G}^T\,{\mathbf W}\,{\mathbf G})^{-1}{\mathbf G}^T\,{\mathbf W}\,{\mathbf Y} \qquad \mbox{(3)}$

${\mathbf P}_{_{\mathbf \Delta X_W}}=({\mathbf G}^T\,{\mathbf W}\,{\mathbf G})^{-1}{\mathbf G}^T \,{\mathbf W}\,\,{\mathbf R}\,\,{\mathbf W}\,{\mathbf G}({\mathbf G}^T\,{\mathbf W}\,{\mathbf G})^{-1} \qquad \mbox{(4)}$

Notes

1. ^ Notice that, if ${\mathbf W}$ is a diagonal matrix, thence, the norm $\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}$, where the terms in the sums are weighted by the diagonal elements $w_i$. Thence, it follows that the measurements associated to large $w_i$ values will contribute more to the adjustment.