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Adjusting a 2D Helmert transformation within a Gauss–Helmert Model with a singular dispersion matrix where BQ is of smaller rank than B

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Abstract

The case of a singular dispersion matrix within the Gauss–Helmert Model has been considered before, most recently even allowing the rank of BQ to be smaller than the rank of B. In this contribution the emphasis is shifted towards an illuminating example, the 2D Helmert transformation.

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Acknowledgments

The first author would like to gratefully acknowledge the support of a Feodor-Lynen Research Fellowship from the Alexander-von-Humboldt Foundation (Germany), and the School of Earth Sciences at the Ohio State University (Columbus/OH, USA), with Prof. Schaffrin as his host. This manuscript was actually completed when the second author visited Prof. Neitzel, again with funds of the AvH-Foundation, which is also gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Geodesy and Geoinformation Science, Technische Universität, Berlin, Germany

    Frank Neitzel

  2. Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus, OH, USA

    Frank Neitzel & Burkhard Schaffrin

Authors
  1. Frank Neitzel
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  2. Burkhard Schaffrin
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Correspondence to Frank Neitzel.

Appendix

Appendix

Estimated dispersion matrix of the estimated parameters \( \hat{\xi }_{0} ,\;\hat{\xi }_{1} ,\;\hat{\xi }_{2} ,\;\hat{\xi }_{3} , \) respectively for \( \hat{\omega } \) and \( \hat{\alpha }, \) from inverting (2.3), including their covariances, for the numerical example:

$$ \begin{aligned} \hat{D}\left\{ {\left[ {\begin{array}{rrrr} {\hat{\xi }_{0} } \\ {\hat{\xi }_{1} } \\ {\hat{\xi }_{2} } \\ {\hat{\xi }_{3} } \\ \end{array} } \right]} \right\} & = \left[ {\begin{array}{*{20}r} {1.673{\text{E}}{-}05} & {1.018{\text{E}}{-}05} & { - 4.469{\text{E}}{-}08} & {7.078{\text{E}}{-}09} \\ {1.018{\text{E}}{-}05} & {6.191{\text{E}}{-}06} & { - 2.718{\text{E}}{-}08} & {4.306{\text{E}}{-}09} \\ { - 4.469{\text{E}}{-}08} & { - 2.718{\text{E}}{-}08} & {1.194{\text{E}}{-}10} & { - 1.891{\text{E}}{-}11} \\ {7.078{\text{E}}{-}09} & {4.306{\text{E}}{-}09} & { - 1.891{\text{E}}{-}11} & {2.994{\text{E}}{-}12} \\ \end{array} } \right], \\ \hat{D}\left\{ {\left[ {\begin{array}{*{20}r} {\hat{\xi }_{0} } \\ {\hat{\xi }_{1} } \\ {\hat{\omega }} \\ {\hat{\alpha }} \\ \end{array} } \right]} \right\} & = \left[ {\begin{array}{rrrr} {1.673{\text{E}}{-}05} & {1.018{\text{E}}{-}05} & { - 4.524{\text{E}}{-}08} & {3.653{\text{E}}{-}15} \\ {1.018{\text{E}}{-}05} & {6.191{\text{E}}{-}06} & { - 2.752{\text{E}}{-}08} & {2.196{\text{E}}{-}15} \\ { - 4.524{\text{E}}{-}08} & { - 2.752{\text{E}}{-}08} & {1.224{\text{E}}{-}10} & { - 9.849{\text{E}}{-}18} \\ {3.653{\text{E}}{-}15} & {2.196{\text{E}}{-}15} & { - 9.849{\text{E}}{-}18} & {5.159{\text{E}}{-}20} \\ \end{array} } \right]. \\ \end{aligned} $$
(A. 1)

Estimated dispersion matrices for the residuals and their cross-covariance matrix:

$$ \begin{aligned} \hat{D}\left\{ {\tilde{e}_{xy} } \right\} = \left[ {\begin{array}{rrrrr} {1.115{\text{E}}{-}06} & { - 1.577{\text{E}}{-}07} & { - 4.157{\text{E}}{-}07} & { - 1.618{\text{E}}{-}07} & { - 3.294{\text{E}}{-}07} \\ { - 1.577{\text{E}}{-}07} & {9.454{\text{E}}{-}07} & { \, 2.903{\text{E}}{-}07} & { - 4.013{\text{E}}{-}07} & {2.116{\text{E}}{-}07} \\ { - 4.157{\text{E}}{-}07} & {2.903E - 07} & {9.542{\text{E}}{-}07} & {6.724{\text{E}}{-}08} & { - 3.993{\text{E}}{-}07} \\ { - 1.618{\text{E}}{-}07} & { - 4.013{\text{E}}{-}07} & {6.724{\text{E}}{-}08} & {9.942{\text{E}}{-}07} & {4.795{\text{E}}{-}08} \\ { - 3.294{\text{E}}{-}07} & {2.116{\text{E}}{-}07} & { - 3.993{\text{E}}{-}07} & {4.795{\text{E}}{-}08} & {1.020{\text{E}}{-}06} \\ { - 2.431{\text{E}}{-}07} & { - 1.294{\text{E}}{-}07} & { - 2.479{\text{E}}{-}07} & { - 6.001{\text{E}}{-}07} & {1.976{\text{E}}{-}07} \\ { - 1.007{\text{E}}{-}08} & { - 1.241{\text{E}}{-}07} & { - 1.048{\text{E}}{-}07} & {2.315{\text{E}}{-}07} & { - 2.805{\text{E}}{-}07} \\ {1.192{\text{E}}{-}07} & { - 4.400{\text{E}}{-}08} & { - 2.542{\text{E}}{-}07} & { - 3.847{\text{E}}{-}08} & { - 3.876{\text{E}}{-}07} \\ { - 3.602{\text{E}}{-}07} & { - 2.201{\text{E}}{-}07} & { - 3.442{\text{E}}{-}08} & { - 1.848{\text{E}}{-}07} & { - 1.069{\text{E}}{-}08} \\ {4.434{\text{E}}{-}07} & { - 3.707{\text{E}}{-}07} & {1.446{\text{E}}{-}07} & {4.564{\text{E}}{-}08} & { - 6.951{\text{E}}{-}08} \\ \end{array} } \right. \hfill \\ \left. {\begin{array}{rrrrr} { - 2.431{\text{E}}{-}07} & { - 1.007{\text{E}}{-}08} & {1.192{\text{E}}{-}07} & { - 3.602{\text{E}}{-}07} & {4.434{\text{E}}{-}07} \\ { - 1.294{\text{E}}{-}07} & { - 1.241{\text{E}} - 07} & { - 4.400{\text{E}}{-}08} & { - 2.201{\text{E}} - 07} & { - 3.707{\text{E}}{-}07} \\ { - 2.479{\text{E}} - 07} & { - 1.048{\text{E}}{-}07} & { - 2.542{\text{E}}{-}07} & { - 3.442{\text{E}}{-}08} & {1.446{\text{E}}{-}07} \\ { - 6.001{\text{E}}{-}07} & {2.315{\text{E}}{-}07} & { - 3.847{\text{E}}{-}08} & { - 1.848{\text{E}}{-}07} & {4.564{\text{E}}{-}08} \\ {1.976{\text{E}}{-}07} & { - 2.805{\text{E}}{-}07} & { - 3.876{\text{E}}{-}07} & { - 1.069{\text{E}}{-}08} & { - 6.951{\text{E}}{-}08} \\ {1.153{\text{E}}{-}06} & {1.913{\text{E}}{-}07} & { - 3.772{\text{E}}{-}07} & {1.020{\text{E}}{-}07} & { - 4.651{\text{E}}{-}08} \\ {1.913{\text{E}}{-}07} & {7.262{\text{E}}{-}07} & {8.462{\text{E}}{-}08} & { - 3.309{\text{E}}{-}07} & { - 3.833{\text{E}}{-}07} \\ { - 3.772{\text{E}}{-}07} & {8.462{\text{E}}{-}08} & {1.010{\text{E}}{-}06} & {4.380{\text{E}}{-}07} & { - 5.506{\text{E}}{-}07} \\ {1.020{\text{E}} - 07} & { - 3.309{\text{E}}{-}07} & {4.380{\text{E}}{-}07} & {7.362{\text{E}}{-}07} & { - 1.351{\text{E}}{-}07} \\ { - 4.651{\text{E}}{-}08} & { - 3.833{\text{E}}{-}07} & { - 5.506{\text{E}}{-}07} & { - 1.351{\text{E}}{-}07} & {9.222{\text{E}}{-}07} \\ \end{array} } \right], \hfill \\ \end{aligned} $$
(A. 2)
$$ \begin{aligned} \hat{D}\left\{ {\tilde{e}_{xy} } \right\} = \left[ {\begin{array}{rrrrr} {2.991{\text{E}}{-}05} & { - 3.190{\text{E}} - 06} & { - 1.122{\text{E}}{-}05} & { - 4.312{\text{E}}{-}06} & { - 8.240{\text{E}}{-}06} \\ { - 3.190{\text{E}}{-}06} & {2.323{\text{E}}{-}05} & {7.346{\text{E}}{-}06} & { - 9.843{\text{E}}{-}06} & {4.679{\text{E}}{-}06} \\ { - 1.122{\text{E}}{-}05} & {7.346{\text{E}}{-}06} & {2.410{\text{E}}{-}05} & {1.490{\text{E}}{-}06} & { - 9.628{\text{E}}{-}06} \\ { - 4.312{\text{E}}{-}06} & { - 9.843{\text{E}}{-}06} & {1.490{\text{E}}{-}06} & {2.614{\text{E}}{-}05} & {2.163{\text{E}}{-}06} \\ { - 8.240{\text{E}}{-}06} & {4.679{\text{E}}{-}06} & { - 9.628{\text{E}}{-}06} & {2.163{\text{E}}{-}06} & {2.481{\text{E}}{-}05} \\ { - 7.043{\text{E}}{-}06} & { - 3.590{\text{E}}{-}06} & { - 5.465{\text{E}}{-}06} & { - 1.614{\text{E}}{-}05} & {4.314{\text{E}}{-}06} \\ { - 2.618{\text{E}}{-}07} & { - 3.061{\text{E}}{-}06} & { - 2.569{\text{E}}{-}06} & {5.718{\text{E}}{-}06} & { - 6.511{\text{E}}{-}06} \\ {3.213{\text{E}}{-}06} & { - 1.133{\text{E}}{-}06} & { - 6.805{\text{E}}{-}06} & { - 1.125{\text{E}}{-}06} & { - 9.485{\text{E}}{-}06} \\ { - 1.018{\text{E}}{-}05} & { - 5.774{\text{E}}{-}06} & { - 6.766{\text{E}}{-}07} & { - 5.058{\text{E}}{-}06} & { - 4.276{\text{E}}{-}07} \\ {1.133{\text{E}}{-}05} & { - 8.664{\text{E}}{-}06} & {3.435{\text{E}}{-}06} & {9.654{\text{E}}{-}07} & { - 1.671{\text{E}}{-}06} \\ \end{array} } \right. \hfill \\ \left. {\begin{array}{rrrrr} { - 7.043{\text{E}}{-}06} & { - 2.618{\text{E}}{-}07} & {3.213{\text{E}}{-}06} & { - 1.018{\text{E}}{-}05} & {1.133{\text{E}}{-}05} \\ { - 3.590{\text{E}}{-}06} & { - 3.061{\text{E}}{-}06} & { - 1.133{\text{E}}{-}06} & { - 5.774{\text{E}}{-}06} & { - 8.664{\text{E}}{-}06} \\ { - 5.465{\text{E}}{-}06} & { - 2.569{\text{E}}{-}06} & { - 6.805{\text{E}}{-}06} & { - 6.766{\text{E}}{-}07} & {3.435{\text{E}}{-}06} \\ { - 1.614{\text{E}}{-}05} & {5.718{\text{E}}{-}06} & { - 1.125{\text{E}}{-}06} & { - 5.058{\text{E}}{-}06} & {9.654{\text{E}}{-}07} \\ { - .314{\text{E}}{-}06} & { - 6.511{\text{E}}{-}06} & { - 9.485{\text{E}}{-}06} & { - 4.276{\text{E}}{-}07} & { - 1.671{\text{E}}{-}06} \\ {3.122{\text{E}}{-}05} & {5.442{\text{E}}{-}06} & { - 1.045{\text{E}}{-}05} & {2.752{\text{E}}{-}06} & { - 1.047{\text{E}}{-}06} \\ {5.442{\text{E}}{-}06} & {1.823{\text{E}}{-}05} & {9.436{\text{E}}{-}07} & { - 8.888{\text{E}}{-}06} & { - 9.043{\text{E}}{-}06} \\ { - 1.045{\text{E}}{-}05} & {9.436{\text{E}}{-}07} & {2.654{\text{E}}{-}05} & {1.213{\text{E}}{-}05} & { - 1.384{\text{E}}{-}05} \\ {2.752{\text{E}}{-}06} & { - 8.888{\text{E}}{-}06} & {1.213{\text{E}}{-}05} & {2.018{\text{E}}{-}05} & { - 4.054{\text{E}}{-}06} \\ { - 1.047{\text{E}}{-}06} & { - 9.043{\text{E}}{-}06} & { - 1.384{\text{E}}{-}05} & { - 4.054{\text{E}}{-}06} & {2.258{\text{E}}{-}05} \\ \end{array} } \right], \hfill \\ \end{aligned} $$
(A. 3)
$$ \begin{aligned} Cov\left\{ {\tilde{e}_{XY} ,\;\tilde{e}_{xy} } \right\} = \left[ {\begin{array}{rrrrr} { - 5.719{\text{E}}{-}06} & { - 9.516{\text{E}}{-}08} & {1.956{\text{E}}{-}06} & {1.142{\text{E}}{-}06} & {1.459{\text{E}}{-}06} \\ {1.542{\text{E}}{-}06} & { - 4.617{\text{E}}{-}06} & { - 1.775{\text{E}}{-}06} & {1.782{\text{E}}{-}06} & { - 1.164{\text{E}}{-}06} \\ {2.316{\text{E}}{-}06} & { - 1.126{\text{E}}{-}06} & { - 4.733{\text{E}}{-}06} & { - 1.095{\text{E}}{-}06} & {1.806{\text{E}}{-}06} \\ {4.929{\text{E}}{-}07} & {2.141{\text{E}}{-}06} & {4.525{\text{E}}{-}07} & { - 5.039{\text{E}}{-}06} & { - 7.172{\text{E}}{-}07} \\ {1.820{\text{E}}{-}06} & { - 7.995{\text{E}}{-}07} & {2.041{\text{E}}{-}06} & {7.664{\text{E}}{-}08} & { - 4.958{\text{E}}{-}06} \\ {1.116{\text{E}}{-}06} & {8.423{\text{E}}{-}07} & {7.664{\text{E}}{-}07} & {3.206{\text{E}}{-}06} & { - 7.501{\text{E}}{-}08} \\ { - 4.803{\text{E}}{-}08} & {6.302{\text{E}}{-}07} & {7.093{\text{E}}{-}07} & { - 1.078{\text{E}}{-}06} & {1.559{\text{E}}{-}06} \\ { - 6.329{\text{E}}{-}07} & {1.260{\text{E}}{-}07} & {1.245{\text{E}}{-}06} & {3.949{\text{E}}{-}07} & {1.644{\text{E}}{-}06} \\ {1.632{\text{E}}{-}06} & {1.390{\text{E}}{-}06} & {2.581{\text{E}}{-}08} & {9.542{\text{E}}{-}07} & {1.347{\text{E}}{-}07} \\ { - 2.518{\text{E}}{-}06} & {1.507{\text{E}}{-}06} & { - 6.890{\text{E}}{-}07} & { - 3.437{\text{E}}{-}07} & {3.117{\text{E}}{-}07} \\ \end{array} } \right. \hfill \\ \left. {\begin{array}{rrrrr} {1.481{\text{E}}{-}06} & {1.452{\text{E}} - 07} & { - 5.900{\text{E}} - 07} & {2.159{\text{E}}{-}06} & { - 1.937{\text{E}}{-}06} \\ {4.812{\text{E}}{-}07} & {5.873{\text{E}}{-}07} & {3.193{\text{E}} - 07} & {8.094{\text{E}} - 07} & {2.034{\text{E}}{-}06} \\ {1.560{\text{E}}{-}06} & {3.235{\text{E}}{-}07} & {1.358{\text{E}}{-}06} & {2.875{\text{E}}{-}07} & { - 6.979{\text{E}}{-}07} \\ {2.971{\text{E}}{-}06} & { - 1.191{\text{E}}{-}06} & {9.159{\text{E}}{-}09} & {9.632{\text{E}}{-}07} & { - 8.203{\text{E}}{-}08} \\ { - 1.801{\text{E}}{-}06} & {1.099{\text{E}}{-}06} & {2.167{\text{E}}{-}06} & { - 1.657{\text{E}}{-}09} & {3.572{\text{E}}{-}07} \\ { - 5.941{\text{E}}{-}06} & { - 1.259{\text{E}}{-}06} & {1.740{\text{E}}{-}06} & { - 5.486{\text{E}}{-}07} & {1.521{\text{E}}{-}07} \\ { - 7.366{\text{E}}{-}07} & { - 3.575{\text{E}}{-}06} & { - 1.001{\text{E}}{-}06} & {1.355{\text{E}}{-}06} & {2.185{\text{E}}{-}06} \\ {2.200{\text{E}}{-}06} & {3.781{\text{E}}{-}07} & { - 5.134{\text{E}}{-}06} & { - 2.634{\text{E}}{-}06} & {2.413{\text{E}}{-}06} \\ { - 5.032{\text{E}}{-}07} & {2.007{\text{E}}{-}06} & { - 1.934{\text{E}}{-}06} & { - 3.800{\text{E}}{-}06} & {9.283{\text{E}}{-}08} \\ {2.885{\text{E}}{-}07} & {1.485{\text{E}}{-}06} & {3.066{\text{E}}{-}06} & {1.410{\text{E}}{-}06} & { - 4.518{\text{E}}{-}06} \\ \end{array} } \right]. \hfill \\ \end{aligned} $$
(A. 4)

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Neitzel, F., Schaffrin, B. Adjusting a 2D Helmert transformation within a Gauss–Helmert Model with a singular dispersion matrix where BQ is of smaller rank than B . Acta Geod Geophys 52, 479–496 (2017). https://doi.org/10.1007/s40328-016-0184-2

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