The joint error covariance matrix of observed and
predicted data for Kalman Filtering (KF) of a typical geophysical
problem is immense but **sparse**.
The total computing load of an exact numerical inversion is
proportional to the cubic of size N of this N*N square symmetric matrix.
Prof. Dr. Tzvi Gal-Chen (1941-1994), the late Professor of Meteorology in the University of
Oklahoma (OU), identified this immense magnitude of a
reliable Kalman solution
and ended up with proposing parallel supercomputing,
see Bull. Am. Met. Soc., Vol. 71, No. 5, May 1990, page
684.

It has turned out that the sparsity of the error covariance matrix can
be exploited **analytically** as first outlined by Prof. F. R. Helmert already in
1880 and later by
Lange in
1987.
This often makes an exact solution of all the parameters to be
estimated up to N times faster to compute.
The patented Fast Kalman Filtering
(FKF,
1990,
1993, and
1996)
exploits the Helmert-Wolf blocking
(HWB, 1978)
method by taking the following steps:

- the large error covariance matrix is block-diagonalized by perfoming a physical and/or statistical analysis for finding a set of Canonical Common Factors that can explain all dependencies between the data blocks to be used (i.e. a physical or statistical explanation has to be found for all those correlation coefficients that lie outside the blocks in the diagonal);
- the Augmented Model of an optimal Kalman Filter (KF) is then made Canonical Block-Angular (CBA) with the help fo those Canonical Common Factors;
- Frobenius inversion formula i.e. Schur complements are applied to the Bordered-Block-Diagonal (BBD) system of Normal Equations; so that,
- the semi-analytic FKF formulas are then obtained for the real-time computations of operational Kalman filtering.

The suggested parallel computing and the patented FKF computations are complementary. Stability of the FKF filtering is crucial for safety of many sophisticated services that will increasingly rely on automation. Estimating various time-varying model and calibration parameters (including the Canonical Common Variates) of any large system calls for their sufficient observability. Continuously inflowing observed and predicted data can then be rapidly analysed in large moving batches for an improved observability. Fortunately, the blockwise computing of FKF speeds up Kalman filtering the more drastically the more data are to be processed at a time. Thus, supercomputing may not always be needed for the Statistical Calibration and Model Identification of Adaptive Kalman Filtering (AKF). In fact, light-weight position-finding (LiteFix) devices are foreseen to be based on FKF for use in most secure operations of safety-critical services.

* Last revised: June 11, 2004