The A-optimal regularization parameter in uniform Tykhonov-Phillips regularization — α-weighted BLE-

  • Jianqing Cai
  • Erik W. Grafarend
  • Burkhard Schaffrin
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 127)


Numerical tests have documented that the estimate \(\hat \xi \) of type BLUUE of the parameter vector ξ within a linear Gauss-Markov model {=E{y}, Σ y = D{y}} is not robust against outliers in the stochastic observation vector y. It is for this reason that we give up the postulate of unbiasedness, but keeping the set-up of a linear estimation \(\hat \xi = Ly\) of homogeneous type. Grafarend and Schaffrin (1993) as well as Schaffrin (2000) have systematically derived the best linear estimators of type homBLE (Best homogeneously Linear Estimation), S-homBLE and α-homBLE of the fixed effects ξ, which turn out to enhance the best linear uniformly unbiased estimator of type BLUUE, but suffer from the effect being biased. Here the regularization parameter in uniform Tykhonov-Phillips regularization (α-weighted BLE) is determined by minimizing the trace of the Mean Square Error matrix MSE α,s {\(\hat \xi \)} (A-optimal design) in the general case. In lieu of a case study, both model and estimators are tested and analyzed with numerical results computed from simulated direct observations of a random tensor of type strain rate.


A-optimal design Tykhonov-Phillips regularization α-homBLE α-HAPS ridge estimator biased estimation Mean Square Error 


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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jianqing Cai
    • 1
  • Erik W. Grafarend
    • 1
  • Burkhard Schaffrin
    • 2
  1. 1.Department of Geodesy and GeoInformaticsUniversity of StuttgartStuttgartGermany
  2. 2.Department of Civil and Environmental Engineering and Geodetic ScienceThe Ohio State UniversityColumbusUSA

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