The FKF formulas

The FKF formulas for computing state parameters repeatedly (t = 1, 2, 3,... ) are obtained by an augmentation from the Helmert-Wolf formulas as follows:

st = (X'tXt)-1X't(Yt - Gtct)   ............... Weighted Averages (WA) of calibrated data
ct = (G'iRiGi)-1G'iRi Yi   ...... Moving Averages (MA) of computed residuals
st = computed state parameters at time t
Yt = vector of normalized observational data, augmented by state parameter estimates or forecasts
Xt = augmented Jacobian matrix for the state parameters
Gt = augmented Jacobian matrix for calibration parameters and/or other common state parameters (e.g. Canonical Common Factors)
ct = the computed calibration parameters and/or other common state parameters (e.g. the Canonical Common Variates)
= summation where the index i runs over several time blocks of data
Ri = I-Xi(X'iXi)-1X'i = residual operator generating an "innovation sequence" for time i.

Error estimates of the state vectors s1, s2,…, st, for example, and of the calibration vector ct are obtained by computing the large covariance matrix:

where, for i = 1, 2,…, t:
Ci = (X'iXi)-1
Di = (X'iXi)-1X'iGi
S = (G'iRiGi)-1
and, in particular,

Cov(st) = Ct + DtSDt'
Cov(ct) = S.

This extension of the Helmert-Wolf formulas to Fast K- Filtering (FKF) was first reported in Lange, Antti A. (1989): "An Algorithmic Approach for Improving and Controlling the Quality of Upper-Air Data." WMO Instruments and Observing Methods Report, No. 35, World Meteorological Organization, Geneva, Switzerland, 1989, see page 5; and,
thereafter published both in Lange, Antti A. (1990a): "Real-time Optimum Calibration of large sensor systems by K- Filtering." IEEE PLANS '90 - Position Location and Navigation Symposium Record, March 20-23, 1990, pp. 146-149; and,
subsequently in Lange, Antti A. (1990b): "Apparatus and method for calibrating a sensor system." International Application Published under the Patent Co-operation Treaty (PCT), World Intellectual Property Organization, International Bureau, WO 90/13794, PCT/FI90/00122, 15 November 1990.
See also Lange (1993 and 1997).

The formula for computing the large covariance matrix above was first published in Lange (1982) “Multipath propagation of VLF Omega signals”, IEEE PLANS '82 - Position Location and Navigation Symposium Record, December 1982, see pages 302-309 (308). The recursive FKF solution requires these error covariances in realtime for weighting different data correctly in various mobile (or kinematic) positioning, navigation and process control applications, see Statistical Calibration of Observing Systems by Lange (1999) or more recent discussions in Lange (2000 or 2003).

A rigorous derivation of all the FKF formulas was published in Lange, A. A. (2001): "Simultaneous Statistical Calibration of the GPS signal delay measurements with related meteorological data", Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, Vol. 26, No. 6-8, pp. 471-473. A derivation of these FKF formulas including the large covariance matrix above is given in blackboard snapshots 1, 2 and 3. The FKF method makes it possible to exploit Rao's MINQUE (Minimun Norm Quadratic Unbiased Estimation) theory for the most reliable operational estimation of accuracies of the state and calibration parameters with smallest low-powered processors.

Sor Etude No. 8
* Last revised February 18, 2005.