By Antti Lange, PhD
In Tomography, the 3-dimensional distribution of e.g. atmospheric water vapor is computed from delay measurements of many crossing satellite signals observed by a dense network of low cost GPS receivers. However, overall absolute levels of the measurements needs to be estimated correctly. The FKF method provides the most effective means for adjusting both the GPS measurements and the related meteorological data simultaneously in real-time according to best possible calibration standards. As the various physical dependencies are non-linear the optimal results can be achieved only by estimating the parameters simultaneously e.g. using an iterative Gauss-Newton process.
However, the computing load of this inversion problem is proportional to the cubic of the data volume that needs simultaneous processing. This requirement has gone beyond manageable limits for many sophisticated systems (Gal-Chen et al., 1990). The Fast computing of K-Filters (FKF) for Optimum Calibration (O/C) of meteorological data (Lange, 1988) applies also to Statistical Calibration (Lange, 1999) of the GPS signal and meteorological data.
The joint regression equation system is initially transformed into a Canonical Block-Angular (CBA) form. All effects stemming from correlated measurement errors can then be taken into account e.g. through the various statistical means outlined in (Lange, 1969). The main advantage of FKF computing is that the enormously large Normal Equation (NEQ) system is thereafter solved semi-analytically, see e.g. Brockmann (1997). Large moving windows of data can now be analyzed for improved estimation of all unknown parameters. Some of them may stay seriously unobserved if a sequential esitmation method based on Square Root filtering is used (Bierman, 1997). The covariance matrix of all the estimated parameters is a by-product and its true absolute magnitude can be determined even in real-time by using the theory of Minimum Norm Quadratic Unbiased Estimation (MINQUE), see Rao (1972).
Some related publications
Extended abstracts presented at the COST Action 716 Workshop in Oslo, 10-12 July 2000.
Bierman, G. J.: Factorization Methods for Discrete Sequential Estimation, Academic Press, San Diego, Calif., 1977.
Brockmann, Elmur: Combination of solutions for geodetic and geodynamic applications of the Global Positioning System (GPS), Geodätisch - geophysikalische Arbeiten in der Schweiz, Volume 55, Schweitzerische Geodätische Kommission, 1997.
Gal-Chen, Tzvi, et al.: Report of the Critical Review Panel – Lower Tropospheric Profiling Symposium: Needs and Technologies, Bull. Am. Met. Soc., Vol. 71, No. 5, 680-687, 1990.
Helmert, F. R.: Die mathematischen und physikalischen Theorien der höheren Geodäsie, 1. Teil, Leipzig, 1880.
Lange, Antti A.: Simultaneous Statistical Calibration of the GPS signal delay measurements with related meteorological data, Phys. Chem. Earth (A), Vol. 26, No. 6-8, pp. 471-473, 2001.
Lange, Antti A.: Statistical Calibration of observing systems, Contributions, No. 22, Finnish Meteorological Institute, Helsinki, Finland, 1999.
Lange, Antti A.: A high-pass filter for Optimum Calibration of observing systems with applications, In: Simulation and optimization of large systems, edited by Andrzej. J. Osiadacz, Clarendon Press, Oxford, 311-327, 1988.
Lange, Antti A.: Kanonisen analyysin laskennoista, Laskentakeskuksen Tiedonantoja, No. 1, Jyväskylän Yliopisto, 1969.
Rao, C. R.: Estimation of variance and covariance components in linear models, J. Am. Stat. Assoc., Vol. 67, No. 337, 112-115, 1972.