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: European patents 0470140 (1993) and 0639261 (1996)
and US patents 5506794 (1996), 5654907 (1997) and 6202033 (2001),
Australia, Barbados, Brazil, Bulgaria,..., Japan, Madagascar, Malawi, OAPI (Africa),
Russia (EAPO), Switzerland, Turkey, etc.

**The FKF Formula**

where

X

G

R

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

This FKF formula above stems from
the semi-analytical inversion method for sparse symmetric matrices
that was described in presentation "A High-pass Filter for Optimum Calibration
of Observing Systems with Applications"
( Lange, 1986), 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.
An academic dissertation
was approved in 1999 by the University of Helsinki, Finland.

**Scope of FKF**

**Provenly stable**computations of optimal K-Filtering with the observability and controllability conditions satisfied can now be applied to largest observing systems like that of the World Weather Watch as 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).

**Reference**

Please have a look also at FKFware. Back to FKF.net

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* Last revised January 16, 2002