Least squares collocation
The collocation according to the smallest squares (according to the Latin collocatio arrangement, common position), engl. least squares collocation , is a combined interpolation and adjustment procedure in which, in contrast to the normal adjustment calculation, data with very different characteristics can be processed.
Whoever developed the method and its basics for the first time has not yet been researched beyond doubt. At the Institute for Computer Engineering of the then TH Dresden, Horst Kadner developed collocation methods in 1958 as part of his doctoral thesis and completed his habilitation in 1966 on approximation methods for linear integration equations of the second kind based on collocation. From 1969 Horst Kadner developed as a full professor for mathematical cybernetics and computing technology at the TU Dresden solution methods for a special class of integral equations on the basis of collocation methods.
At the end of the 1970s, these methods were adopted by geodesist and mathematician Helmut Moritz (Berlin / Graz) for the purpose of integrated geoid determination in order to be able to process geometric and physical data of the earth's shape and the earth's gravity field in one pour. Moritz also gave solutions to the collocation problem and the covariance matrix in steps in order to reduce the computer computing times with large amounts of data.
Extensive applications come from Hans Sünkel (integrated local geoid determination ) and Christian Tscherning (regional gravimetry). The first astro-geodetic geoid determination using LSC took place in 1982 at Graz University of Technology . She was able to increase the accuracy of the Austrian astrogeoid (on average ± 6 cm from 700 measurement points of the vertical deviation ) by including a global harmonic gravity model (RHRapp, up to 180th order) by around a quarter and isolate a local data error. Three years later, the inclusion of around 10,000 gravity anomalies increased accuracy to ± 4 cm.
Since around 1990, the collocation has also served as the basis for large-scale gravity field modeling including the development of spherical functions in satellite geodesy , including in two program systems at German universities, GRAVSOFT and Opera (geodesy) . Applications in Northern Europe (Tscherning & Forsberg 1986–1993), in Italy, Spain (Simo, Catalao & Sevilla 1994) and in Turkey (Ayhan 1993) showed the advantages of integral calculations by increasing the accuracy of about a third compared to individual solutions.
The specialty of these applications is the minimization of the mean error of the measurements used, in that all data configurations are mapped into one another by rotating the geocenter (hence the name collocation ).
The collocation method is now also used in chemical thermodynamics .
literature
- ZAMM-Journal of Applied Mathematics and Mechanics, Vol. 40, Issue 1–3, Pages 99–113, investigations on the collocation method , Dr. rer. nat. Horst Kadner, from: Dissertation v. 17th February 1958
- In memoriam Horst Kadner , Dresdner UniversitätsJournal, Volume 16, No. 12, July 5, 2005, p. 4
- H. Moritz: Advanced Physical Geodesy . De Gruyter-Verlag, 1980
- GP Bottoni, R.Barzaghi: Fast Collocation . Bulletin Geodesique , Vol. 67, pp. 119-126, Paris 1993
- CC Tscherning: Local Approximation of the Gravity Potential by Least Squares Collocation . KP Schwarz (Ed.): Proceedings of the International Summer School on Local Gravity Field Approximation, Beijing (China), August 21 - September 4, 1984. Publ. 60003, Univ. of Calgary (Canada), pp. 277-362, Calgary 1985
- H. Sünkel et al .: The Geoid in Austria, calculations at TU Graz . In K.Rinner (ed.): Geodät.Arbeiten Österreichs für die Internationale Erdkunde Volume III, pp. 117–146, Graz 1983