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:

= summation where the index i runs over several time blocks of data

Error estimates of the state vectors **s**_{1}, ** s**_{2},…, ** s**_{t},
for example, and of the calibration vector **c**_{t} are obtained by computing the large covariance matrix:

D

S = (

Cov(**s**_{t}) = C_{t} + D_{t}SD_{t}'

Cov(**c**_{t}) = 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.