Helmert–Wolf blocking

The Helmert–Wolf blocking[1] (HWB) is a least squares solution method[2] for a sparse canonical block-angular[3] (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][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. 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. 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. "The Wolf formulas". 9 June 2004. Retrieved 6 June 2017.
  6. http://www.fkf.net/Wolf.jpg
  7. Strang, Gilbert; Borre, Kai (1997). Linear algebra, geodesy, and GPS. Wellesley: Wellesley-Cambridge Press. pp. 507-508. ISBN 9780961408862.


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