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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



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:



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  can be seen as an equilibrium solution.