The Wolf formulas

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The formulas for computing adjustments of all state parameters by the method of Helmert-Wolf blocking (HWB) are as follows:

bi = (X'iXi)-1X'i(yi - Gic) .......... i.e. weighted averages for adjusted data block i
c = (G'i Ri Gi)-1G'iRi yi .... i.e. weighted averages of all computed residuals
bi = vector of the state parameter adjustments for block i of observations
c = vector of the common adjustments
Xi = Jacobian matrix for the state parameters for block i
Gi = Jacobian matrix for the common adjustments for block i
yi = vector of the observations for block i
= summation where the index i runs over all blocks of observations
Ri = I-Xi(X'iXi)-1X'i = residual operator for block i.

These Wolf's formulas above yield the Best Linear Unbiased Estimates (BLUE) as the error covariance matrices of all the data blocks i can always be transformed into identity matrices without loosing any generality, see Lange's FKF.

Prof. Dr.-Ing. Dr. h.c. mult. Helmut Wolf (1910-1994) was the first to publish these Helmert-Wolf blocking (HWB) formulas that are based on Matrix Calculus, see his paper Wolf, H. (1978): "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 24-28, 1978, pages 319-326. Helmut Wolf was born on May 2, 1910 in Werdau, Sachsen. He worked as full professor and Head of the Institute for Theoretical Geodesy at the University of Bonn since 1955 and died on June 6, 1994.

The error covariance matrix of these HWB adjustments was first published only until in Lange, Antti A. (1982): “Multipath propagation of VLF Omega signals”, IEEE PLANS '82 - Position Location and Navigation Symposium Record, December 1982, see pages 302-309 (308). This semi-analytic inversion method was thereafter described in Lange's two conference presentations under same title "A High-pass Filter for Optimum Calibration of Observing Systems with Applications" at the conference of the Institute of Mathematics and its Applications (IMA) held at the University of Reading, UK, in September 1986, and, at the AMS Sixth Symposium on Meterological Observations and Instrumentation, Jan. 12-16 1987, New Orleans, La, see pages 471, 472, 473, and 474 of the Preprint Volume. Reference is made to the Canonical Block-Angular (CBA) equation system that is described on pages 12-14 and 311-327 of SIMULATION AND OPTIMIZATION OF LARGE SYSTEMS edited by Andrzej J. Osiadacz and published by Clarendon Press/Oxford University Press, Oxford, UK in 1988.

Other derivations and/or uses of these formulas above have been reported e.g. by:
- Antti Lange (1975): “TULU”-projektin loppuraportti. Kriisiajan sääpalvelun kehittämistoimikunta. MATINE, Helsinki, 1975;
- Richard D. Ray (1995), ACM/TOMS Algorithm 471: Least Squares Solution of a Linear Bordered, Block-Diagonal System of Equations, ACM Trans. Math. Softw., 221, No. 1, see pages 20-25;
- Gilbert Strang and Kai Borre (1997) Linear Algebra, Geodesy, and GPS, Wellesley-Cambridge Press, see page 508;
- Elmar Brockmann (1997) Combination of Solutions for Geodetic and Geodynamic Applications of the Global Positioning System (GPS) published in: Geodätisch - geophysikalische Arbeiten in der Schweiz, Volume 55, Schweitzerische Geodätische Kommission, see pages 22-23;
- Antti Lange (1999), "lectio praecursoria", blackboard snapshots 1, 2 and 3; and, (2001), "Simultaneous Statistical Calibration of the GPS signal delay measurements with related meteorological data", Phys. Chem. Earth (A), Vol. 26, No. 6-8, see pages 471-473.

A FORTRAN77 subroutine package for solving these HWB adjustments and their error covariances can be retrieved from callib.for.

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* Last revised June 9, 2004.