# Controlling the Bias Within Free Geodetic Networks

- 45 Downloads

## 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 \(S_k := \operatorname {\mathrm {Diag}}(1,{\ldots },1,0,{\ldots },0)\) 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 \( \operatorname {\mathrm {rk}}(S_k N) = \operatorname {\mathrm {rk}}(N) = \operatorname {\mathrm {rk}}(A) =: q\) 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## References

- Caspary W (2000) Concepts of network and deformation analysis. Monograph 11, School of Geomatic Engineering, University of New South WalesGoogle Scholar
- Grafarend E, Sansò F (eds) (1985) Optimization and design of geodetic networks. Springer, BerlinGoogle Scholar
- Grafarend E, Schaffrin B (1974) Unbiased free net adjustment. Surv Rev 22:200–218CrossRefGoogle Scholar
- Schaffrin B (1985) Network design. In: Grafarend EW, Sansó F (eds) Optimization and design of geodetic networks. Springer, Berlin, pp 548–597CrossRefGoogle Scholar
- Schaffrin B (2013/2014) Modern adjustment computations: a model-based approach, Slides prepared for the IAG Summer School in Taiwan and a Summer Course at Tech. University, Graz/AustriaGoogle Scholar
- Schaffrin B, Iz HB (2002) BLIMPBE and its geodetic applications. In: Adam J, Schwarz K (eds) Vistas for geodesy in the new millenium. In: Springer IAG-symposium, vol 125. Springer, Berlin, pp 377–381CrossRefGoogle Scholar
- Snow K, Schaffrin B (2007) GPS network analysis with BLIMPBE: an alternative to least-squares adjustment for better bias control. J Surv Eng 133(3):114–122CrossRefGoogle Scholar