Helmert-Wolf blocking

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The Helmert-Wolf blocking¹ (HWB) is a least squares solution method for a sparse canonical block-angular (CBA) system of linear equations. Friedrich Robert Helmert (1843-1917) reported on the use of such systems for geodesy in his book "Die mathematischen und physikalischen Theorieen der höheren Geodäsie, 1. Teil" published in Leipzig, 1880. Helmut Wolf (1910-1994) published his direct semianalytic solution² based on ordinary Gaussian elimination in matrix form³ in his paper "The Helmert block method, its origin and development", Proceedings of the Second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks, Arlington, Va. April 24-April 28, 1978, pages 319-326. The HWB solution is really fast to compute but it is optimal only if observational errors do not correlate between the data blocks. Fortunately, 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.

[edit] Notes

Note 1: see GPScom Software Documentation from Geoscience Research Division of NOAA.

Note 2: see formulas (15.56-58) on pages 507-508 of Strang, G. and Borre, K. (1997): Linear Algebra, Geodesy, and GPS, Wellesley-Cambridge Press.

Note 3: see the HWB formula (unnumbered) at the end of page 508 of Strang, G. and Borre, K. (1997): Linear Algebra, Geodesy, and GPS, Wellesley-Cambridge Press.

[edit] Applications

The HWB method is a "must" in 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 (1920- ).