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|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
If such errors () are explicitly written, thence the lineal model is as follows:
where the error term is only known from some statistical properties, usually the mean and covariance matrix .
Due to the error term , in general 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 that minimises the discrepancy in the equations system. That is, the vector providing the "best fit" of in a given sense.
A common criterion used in GNSS is the Least-Squares adjustment, which is defined by the condition:
The discrepancy vector between the measurements and the fitted model is usually called the residual vector:
Thence, the Least-Squares estimator solution defined by equation (3), gives the vector that minimises [footnotes 1] the residuals quadratic norm .
From basic results of linear algebra, it follows that the solution fulfilling the condition (3) is given by:
Substituting (5) and (2) in (4) the post-fit -residual vector is:
where is a symmetrical, idempotent Projection matrix
From (5) and (2) the estimator error can be written as:
Assuming that the measurements minus model (i.e., prefit-residuals) have mean zero errors () and covariance matrix , thence, the mean error, covariance matrix and Mean-Square Error (MSE) of the estimator are given by:
The expression of become simpler by assuming uncorrelated values with identical variance . That is, taking , thence:
For more information, please go to the following articles:
- ^ 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 can be seen as an equilibrium solution.