Controlling the Bias Within Free Geodetic Networks

It is well known that the MInimum NOrm LEast-Squares Solution (MINOLESS) minimizes the bias uniformly since it coincides with the BLUMBE (Best Linear Uniformly Minimum Biased Estimate) in a rank-deficient Gauss-Markov Model as typically employed for free geodetic network analyses. Nonetheless, more often than not, the partial-MINOLESS is preferred where a selection matrix Sk:=Diag(1,…,1,0,…,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$S_k := \operatorname {\mathrm {Diag}}(1,{\ldots },1,0,{\ldots },0)$$ \end{document} is used to only minimize the first k components of the solution vector, thus resulting in larger biases than frequently desired. As an alternative, the Best LInear Minimum Partially Biased Estimate (BLIMPBE) may be considered, which coincides with the partial-MINOLESS as long as the rank condition rk(SkN)=rk(N)=rk(A)=:q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \operatorname {\mathrm {rk}}(S_k N) = \operatorname {\mathrm {rk}}(N) = \operatorname {\mathrm {rk}}(A) =: q$$ \end{document} holds true, where N and A are the normal equation and observation equation matrices, respectively. Here, we are interested in studying the bias divergence when this rank condition is violated, due to q > k ≥ m − q, with m as the number of all parameters. To the best of our knowledge, this case has not been studied before.