FKF-IN-BRIEF

FKF in Nutshell

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.

The FKF Formula

The vector s_t of state parameters at time t is to be computed as follows:

where
y_t= normalized data vector at time t, augmented by state parameter prediction
X_t= augmented Jacobian matrix for the state parameters
G_t= augmented Jacobian matrix for calibration parameters
R_i= I-X_i(X'_iX_i)^-1X'_i= residual operator generating "innovation" sequences
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

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).

References

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 October 28, 2004

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