View source for Helmert–Wolf blocking

The '''Helmert–Wolf blocking''' ('''HWB''') is a [[least squares]] solution method for the solution of a [[sparse matrix|sparse]] [[block matrix|block]] [[system of linear equations]].<ref>{{cite web|url=http://www.ngs.noaa.gov/GRD/GPS/DOC/gpscom/mca.html|title=Making Combined Adjustments|first=Bill|last=Dillinger|date=4 March 1999|access-date=6 June 2017}}</ref> It was first reported by [[Friedrich Robert Helmert|F. R. Helmert]] for use in [[geodesy]] problems in 1880;<ref>{{cite book|first=Friedrich Robert|last=Helmert|title=Die mathematischen und physikalischen Theorien der höheren Geodäsie, 1. Teil|location=Leipzig|date=1880}}</ref> {{ill|Helmut Wolf (geodesist)|de|Helmut Wolf (Geodät)|lt=H. Wolf}} (1910–1994) published his direct semianalytic solution in 1978.<ref name="wolf-web">{{cite web|url=http://www.fkf.net/Wolf.html|title=The Wolf formulas|date=9 June 2004|accessdate=6 June 2017}}</ref><ref name="wolf">{{cite conference |url= https://archive.org/details/proceedingsofsec00inte|title=The Helmert block method—its origins and development|last1= Wolf|first1= Helmut|date= April 1978|publisher= U.S. Dept. of Commerce|book-title= Proceedings of the second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks|pages= 319–326|location= Arlington, Virginia|conference=International Symposium on Problems Related to the Redefinition of North American Geodetic Networks}}</ref>
It is based on ordinary [[Gaussian elimination]] in [[matrix (mathematics)|matrix]] form<ref name="strang">{{cite book|title=Linear algebra, geodesy, and GPS|url=https://archive.org/details/linearalgebrageo00borr|url-access=limited|first1=Gilbert|last1=Strang|first2=Kai|last2=Borre|date=1997|publisher=Wellesley-Cambridge Press|location=Wellesley|isbn=9780961408862|pages=[https://archive.org/details/linearalgebrageo00borr/page/n265 507]-508}}</ref> or partial [[Mathematical optimization|minimization]] form.<ref name="Leick Rapoport Tatarnikov 2015 p. 673">{{cite book | last1=Leick | first1=A. | last2=Rapoport | first2=L. | last3=Tatarnikov | first3=D. | title=GPS Satellite Surveying | publisher=Wiley | year=2015 | isbn=978-1-119-01828-5 | url=https://books.google.com/books?id=FQlhBgAAQBAJ&pg=PA673 | access-date=2022-01-30 | page=673}}</ref>
== Description ==
{{stub section|date=May 2015}}
{{further|Invertible matrix#Blockwise inversion|Block matrix#Inversion}}
{{see also|Fast Kalman Filter#Description|Simultaneous_equations_model#Structural_and_reduced_form}}
== 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|date=May 2015}} The HWB method can be extended to [[fast Kalman filter]]ing (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 ==
* [[Block matrix]]
== Notes ==
{{Reflist}}
{{DEFAULTSORT:Helmert-Wolf blocking}}
[[Category:Statistical algorithms]]
[[Category:Least squares]]
[[Category:Geodesy]]
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