It was understood that **optimal**
K-Filters of
large systems were computationally intractable because of the immense
sizes of their observations and forecast error covariance matrices.
Thus, Dr. T. Gal-Chen, Professor of Meteorology at the University of Oklahoma,
indicated in 1988 that **"1000 CRAYs"** would have to work in tandem for
inverting
these matrices.
Fortunately, Wolf's semi-analytical inversion based on
Helmert's (1880)
blocking method is so effective that the
Fast K-Filter (FKF) computations
can be made as close to the optimal as necessary for
a large number of realtime operational
applications.
Patents have been granted worldwide: Canadian patent 2,236,757 (2004), European patents 0470140 (1993), 0639261 (1996) and 0862730 (2003),
and, US patents 5506794 (1996), 5654907 (1997) and 6202033 (2001), and
Albania, Austria, Australia, Belgium, Brazil, Bulgaria, China, Creece, Denmark, Estonia, Finland,
France, Georgia, Germany, Great Britain, Hong Kong, Italy,... ,
Japan, Korea, Latvia, Lithuania, Luxemburg, Madagascar, Monaco, Norway, OAPI (Africa), Poland, Portugal, Romania (2003),
Russia (EAPO Pat Nro. 001188, 2002), Singapore, Slovenia, Slovakia, Spain, Sweden, Switzerland,
The Netherlands, Turkey, Ukraine, Vietnam, etc.

New patents are pending under
PCT/FI2007/00052.

**The FKF Formula**

where

X

G

R

i= summation index running over long time series of data.

This FKF formula stems from
the Helmert-Wolf semi-analytical inversion method for sparse symmetric matrices and
it was **first** presented at the University of Reading, England, in 1986 in paper
"A High-pass Filter for Optimum Calibration
of Observing Systems with Applications" by Lange,
see 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.
The necessary and sufficient conditions for its numerical stability in realtime applications were
**finally** discovered in 1989 and subsequently disclosed in the FKF patents.

**Scope of FKF**

**The proofably stable**computations of optimal K-Filtering with the observability and controllability conditions satisfied can now be applied to largest observing networks like those of the World Weather Watch (WWW) and Global Navigation Satellite Systems (GNSS) well as to the smallest lightweight position finding devices of ordinary citizens;- All sorts of information including remote sensing data
from radars, lidars, sonars, satellites etc. and simulation models like those used for
Numerical Weather Prediction (NWP) can be combined into forecasts with
**best possible accuracy**; **Objective accuracy estimates**based on Rao's MINQUE theory are attached to the position finding results or the different forecasts for optimal control and decision making purposes;- Most sophisticated hybrid observing systems with
**built-in calibration**can be materialized using the same rigorous K-Filtering Theory as operational navigation receivers and autopilots do today; and, - Long moving windows of data can be used for making these filtering processes to
**learn from their own mistakes**through whitening "innovation" sequences of residuals (e.g. subgrid processes of NWP) by Adaptive K-Filtering (AKF).

**References**

Lange, A. A. (2007): "Fast Kalman Processing of Carrier-Phase Signals from Global Navigation Satellite Systems for Water Vapor Tomography ", Proceedings of the 1st GALILEO Scientific Colloquium, 1-4 October 2007, Toulouse, France.

Lange, A. A. (2003): "Optimal Kalman Filtering for ultra-reliable Tracking", Proceedings of the Symposium on Atmospheric Remote Sensing using Satellite Navigation Systems, 13-15 October 2003, Matera, Italy.

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.

Lange, A. A. (1999): "Statistical Calibration of Observing Systems", Academic Dissertation, Finnish Meteorological Institute Contributions Nro. 22, Helsinki, Finland.

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* Last revised May 19, 2009