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: 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

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 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

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.

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* Last revised January 16, 2002