Helmert–Wolf blocking

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The Helmert–Wolf blocking[1] (HWB) is a least squares solution method[2] for a sparse canonical block-angular[3][better source needed] (CBA) system of linear equations. F. R. Helmert (1843–1917) reported on the use of such systems for geodesy in 1880.[4] H. Wolf (1910–1994)[5] published his direct semianalytic solution[5][6][better source needed][7] based on ordinary Gaussian elimination in matrix form [7] in 1978.[2]

Description[]

Further information: Invertible matrix § Blockwise inversion, and Block matrix § Inversion
See also: Fast Kalman Filter § Description

Limitations[]

The HWB solution is very fast to compute but it is optimal only if observational errors do not correlate between the data blocks. The generalized canonical correlation analysis (gCCA) is the statistical method of choice for making those harmful cross-covariances vanish. This may, however, become quite tedious depending on the nature of the problem.

Applications[]

The HWB method is critical to satellite geodesy and similar large problems.[citation needed] The HWB method can be extended to fast Kalman filtering (FKF) by augmenting its linear regression equation system to take into account information from numerical forecasts, physical constraints and other ancillary data sources that are available in realtime. Operational accuracies can then be computed reliably from the theory of minimum-norm quadratic unbiased estimation (Minque) of C. R. Rao.

See also[]

Notes[]

  1. ^ Dillinger, Bill (4 March 1999). "Making Combined Adjustments". Retrieved 6 June 2017.
  2. ^ Jump up to: a b Wolf, Helmut (April 1978). "The Helmert block method—its origins and development". Proceedings of the second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks. International Symposium on Problems Related to the Redefinition of North American Geodetic Networks. Arlington, Virginia: U.S. Dept. of Commerce. pp. 319–326.
  3. ^ http://fkf.net/equations.gif
  4. ^ Helmert, Friedrich Robert (1880). Die mathematischen und physikalischen Theorien der höheren Geodäsie, 1. Teil. Leipzig.
  5. ^ Jump up to: a b "The Wolf formulas". 9 June 2004. Retrieved 6 June 2017.
  6. ^ http://www.fkf.net/Wolf.jpg
  7. ^ Jump up to: a b Strang, Gilbert; Borre, Kai (1997). Linear algebra, geodesy, and GPS. Wellesley: Wellesley-Cambridge Press. pp. 507-508. ISBN 9780961408862.


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