113:. However, these latter methods may solve the large matrix of all the error variances and covariances only approximately and the data fusion would not be performed in a strictly optimal fashion. Consequently, the long-term stability of Kalman filtering becomes uncertain even if Kalman's observability and controllability conditions were permanently satisfied.
477:
621:(NWP) system can now forecast observations with confidence intervals and their operational quality control can thus be improved. A sudden increase of uncertainty in predicting observations would indicate that important observations are missing (observability problem) or an unpredictable change of weather is taking place (controllability problem).
629:
The computational advantage of FKF is marginal for applications using only small amounts of data in real-time. Therefore, improved built-in calibration and data communication infrastructures need to be developed first and introduced to public use before personal gadgets and machine-to-machine devices
53:
conditions are not continuously satisfied. These conditions are very challenging to maintain for any larger system. This means that even optimal Kalman filters may start diverging towards false solutions. Fortunately, the stability of an optimal Kalman filter can be controlled by monitoring its error
66:
Calibration parameters are a typical example of those state parameters that may create serious observability problems if a narrow window of data (i.e. too few measurements) is continuously used by a Kalman filter. Observing instruments onboard orbiting satellites gives an example of optimal Kalman
593:
This is the FKF method that may make it computationally possible to estimate a much larger number of state and calibration parameters than an ordinary Kalman recursion can do. Their operational accuracies may also be reliably estimated from the theory of
Minimum-Norm Quadratic Unbiased Estimation
44:
Kalman filters are an important filtering technique for building fault-tolerance into a wide range of systems, including real-time imaging. The ordinary Kalman filter is an optimal filtering algorithm for linear systems. However, an optimal Kalman filter is not stable (i.e. reliable) if Kalman's
87:
Even when many measurements are processed simultaneously, it is not unusual that the linearized equation system becomes sparse, because some measurements turn out to be independent of some state or calibration parameters. In problems of
Satellite Geodesy, the computing load of the HWB (and FKF)
83:
of the number of the measurements processed simultaneously. This number can always be set to 1 by processing each scalar measurement independently and (if necessary) performing a simple pre-filtering algorithm to de-correlate these measurements. However, for any large and complex system this
109:. Its coefficient matrix is usually sparse and the exact solution of all the estimated parameters can be computed by using the HWB (and FKF) method. The optimal solution may also be obtained by Gauss elimination using other sparse-matrix techniques or some iterative methods based e.g. on
142:
84:
pre-filtering may need the HWB computing. Any continued use of a too narrow window of input data weakens observability of the calibration parameters and, in the long run, this may lead to serious controllability problems totally unacceptable in safety-critical applications.
96:
Reliable operational Kalman filtering requires continuous fusion of data in real-time. Its optimality depends essentially on the use of exact variances and covariances between all measurements and the estimated state and calibration parameters. This large error
124:
The sparse coefficient matrix to be inverted may often have either a bordered block- or band-diagonal (BBD) structure. If it is band-diagonal it can be transformed into a block-diagonal form e.g. by means of a generalized
Canonical Correlation Analysis
625:
and imaging from satellites are partly based on forecasted information. Controlling stability of the feedback between these forecasts and the satellite images requires a sensor fusion technique that is both fast and robust, which the FKF fulfills.
67:
filtering where their calibration is done indirectly on ground. There may also exist other state parameters that are hardly or not at all observable if too small samples of data are processed at a time by any sort of a Kalman filter.
472:{\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}={\begin{bmatrix}A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1}&-A^{-1}B(D-CA^{-1}B)^{-1}\\-(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{bmatrix}}}
717:
614:(RTK) surveying, mobile positioning and ultra-reliable navigation. First important applications will be real-time optimum calibration of global observing systems in Meteorology, Geophysics, Astronomy etc.
58:). Their precise computation is, however, much more demanding than the optimal Kalman filtering itself. The FKF computing method often provides the required speed-up also in this respect.
563:
724:
510:
587:
88:
method is roughly proportional to the square of the total number of the state and calibration parameters only and not of the measurements that are billions.
121:
The Fast Kalman filter applies only to systems with sparse matrices, since HWB is an inversion method to solve sparse linear equations (Wolf, 1978).
80:
821:
Lange, Antti (2001). "Simultaneous
Statistical Calibration of the GPS signal delay measurements with related meteorological data".
36:
navigation up to the centimeter-level of accuracy and satellite imaging of the Earth including atmospheric tomography.
805:
126:
903:. Simulation and Optimization of Large Systems. Oxford: Oxford University Press/Clarendon Press. pp. 311–327.
106:
718:
Star
Tracker/Gyro Calibration and Attitude Reconstruction for the Scientific Satellite Odin - In Flight Results
618:
769:"Combination of Solutions for Geodetic and Geodynamic Applications of the Global Positioning System (GPS)"
133:
610:
The FKF method extends the very high accuracies of
Satellite Geodesy to Virtual Reference Station (VRS)
132:
Such a large matrix can thus be most effectively inverted in a blockwise manner by using the following
963:
483:
21:
517:
968:
941:
There are other Fast Kalman
Algorithms designed for special signal processing purposes, see e.g.
742:
691:
110:
942:
8:
611:
492:
647:
572:
917:
834:
838:
801:
663:
Kalman, Rudolf (1960). "A New
Approach to Linear Filtering and Prediction Problems".
98:
830:
672:
102:
897:
A high-pass filter for
Optimum Calibration of observing systems with applications
768:
76:
50:
948:
883:. Atmospheric Remote Sensing using Satellite Navigation Systems. Matera, Italy.
622:
857:
602:
and used for controlling the stability of this optimal fast Kalman filtering.
957:
842:
46:
29:
716:
Jacobsson, B; Nylund, M; Olssoon, T; Vandermarcq, O; Vinterhav, E (2001).
566:
935:
858:
Using
Helmert–Wolf blocking for diagnosis & treatment of GNSS errors
512:
a large block- or band-diagonal (BD) matrix to be easily inverted, and,
863:(Report). Bordeaux: 22nd ITS World Congress. Technical PAper ITS-1636.
676:
923:
599:
875:
823:
Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy
649:
Making Combined Adjustments [GPScom Software Documentation]
895:
715:
25:
929:
595:
55:
20:, devised by Antti Lange (born 1941), is an extension of the
54:
variances if only these can be reliably estimated (e.g. by
33:
652:(Technical report). Geoscience Research Division of NOAA.
200:
152:
575:
520:
495:
145:
949:
Kalman filter recipes for real-time image processing
877:
Optimal Kalman Filtering for ultra-reliable Tracking
776:
Geodaetisch-geophysikalische Arbeiten in der Schweiz
581:
557:
504:
471:
955:
894:Lange, Antti (1988). Andrez J. Osiadacz (ed.).
800:. Wellesley-Cambridge Press. pp. 507–508.
750:Finnish Meteorological Institute Contributions
743:"Statistical Calibration of Observing Systems"
699:Finnish Meteorological Institute Contributions
692:"Statistical Calibration of Observing Systems"
829:(6–8). Amsterdam: Elsevier Science: 471–473.
28:to safety-critical real-time applications of
795:
766:
796:Strange, Gilbert; Borre, Borre (1997).
956:
943:Stabilizing the Fast Kalman Algorithms
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789:
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79:of an ordinary Kalman recursion is
13:
936:The error covariance matrix of FKF
70:
14:
980:
911:
565:a much smaller matrix called the
874:Lange, Antti (15 October 2003).
798:Linear Algebra, Geodesy, and GPS
81:roughly proportional to the cube
867:
856:Lange, Antti (9 October 2015).
605:
849:
814:
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630:can make the best out of FKF.
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105:from the respective system of
1:
835:10.1016/S1464-1895(01)00086-2
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558:{\displaystyle (D-CA^{-1}B)=}
39:
665:Journal of Basic Engineering
619:Numerical Weather Prediction
7:
10:
985:
134:analytic inversion formula
75:The computing load of the
767:Brockman, Elmar (1997).
723:(Report). Archived from
18:fast Kalman filter (FKF)
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559:
506:
473:
741:Lange, Antti (2008).
690:Lange, Antti (2008).
584:
560:
507:
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22:Helmert–Wolf blocking
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111:Variational Calculus
778:(in Swiss German).
730:on 11 January 2007.
612:Real Time Kinematic
62:Optimum calibration
579:
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505:{\displaystyle A=}
502:
469:
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24:(HWB) method from
964:Signal estimation
677:10.1115/1.3662552
582:{\displaystyle A}
99:covariance matrix
92:Reliable solution
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107:Normal Equations
103:matrix inversion
30:Kalman filtering
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71:Inverse problem
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51:controllability
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969:Linear filters
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623:Remote sensing
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47:observability
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32:(KF) such as
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27:
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725:the original
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671:(1): 34–45.
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606:Applications
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95:
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65:
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17:
15:
117:Description
958:Categories
938:- formulas
932:- formulas
926:- formulas
920:- software
634:References
40:Motivation
843:1464-1895
600:C. R. Rao
539:−
528:−
484:Frobenius
454:−
438:−
427:−
411:−
395:−
379:−
368:−
359:−
347:−
331:−
320:−
303:−
295:−
285:−
269:−
253:−
242:−
225:−
209:−
185:−
756:: 12–13.
705:: 34–45.
486:where
26:geodesy
841:
804:
596:MINQUE
127:(gCCA)
56:MINQUE
901:(PDF)
881:(PDF)
861:(PDF)
772:(PDF)
746:(PDF)
728:(PDF)
721:(PDF)
695:(PDF)
598:) of
567:Schur
839:ISSN
802:ISBN
49:and
34:GNSS
16:The
930:HWB
924:FKF
918:BBD
831:doi
673:doi
482:of
960::
837:.
827:26
825:.
788:^
780:55
774:.
754:22
752:.
748:.
703:22
701:.
697:.
669:82
667:.
136::
129:.
845:.
833::
810:.
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679:.
675::
594:(
589:.
577:A
553:=
550:)
547:B
542:1
535:A
531:C
525:D
522:(
500:=
497:A
465:]
457:1
450:)
446:B
441:1
434:A
430:C
424:D
421:(
414:1
407:A
403:C
398:1
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387:B
382:1
375:A
371:C
365:D
362:(
350:1
343:)
339:B
334:1
327:A
323:C
317:D
314:(
311:B
306:1
299:A
288:1
281:A
277:C
272:1
265:)
261:B
256:1
249:A
245:C
239:D
236:(
233:B
228:1
221:A
217:+
212:1
205:A
198:[
193:=
188:1
179:]
173:D
168:C
161:B
156:A
150:[
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