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 FormulaThe vector st of state parameters at time t is to be computed as follows:
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
ReferenceLange, 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.