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
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Abstract
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 WD Diag.1; : : :; 1; 0; : : :; 0/ 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.Sk N / D rk.N / D rk.A/ DW q 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. Keywords
Bias control Datum deficiency Free geodetic networks
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Introduction
It has long been recognized that the adjustment of free geodetic networks will lead to biased point coordinate estimates when using a rank-deficient Gauss-Markov Model (GMM). Only quantities that can be expressed as functions of the observables may result in unbiased estimates after a (weighted) least-squares adjustment. For more details, we refer to Grafarend and Schaffrin (1974). A far wider scope is reflected in the Springer book Optimization and Design of Geodetic Networks, edited by Grafarend and Sansó (1985), which represents the status of research at the time. For the discussion in the following B. Schaffrin Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus, OH, USA e-mail: [email protected] K. Snow () Topcon Positioning Systems, Inc., Columbus, OH, USA
contribution, most relevant should be the paper “Network Design” by Schaffrin (1985) that can be found in that volume. In particular, estimates of type MINOLESS (MInimum NOrm LEeast-Squares Solution), BLESS (Best LeastSquares Solution), and BLUMBE (Best Linear Uniformly Minimum Biased Estimate) are derived therein and given various representations, including proofs that show their identity under certain conditions. On the other hand, for many practical applications (e.g., Caspary 2000), the general MINOLESS is being replaced by partial MINOLESS for which only the Sk -weighted norm of the estimated parameter vector is minimized, where Sk :D Diag.1; : : : ; 1; 0; : : : ; 0/ is called a “selection matrix.” Unfortunately, Snow and Schaffrin (2007) would then show that the (full) MINOLESS is the only Least-Squares Solution that minimizes the bias uniformly so that the bias of the partial MINOLESS must be monitored separately. Moreover, when Schaffrin and Iz (2002) began studying estimators that minimize the bias at least partially, it turned out that the best among them (BLIMPBE
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