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The Journal of Committee on Geodesy of Polish Academy of Sciences

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A priori noise and regularization in least squares collocation of gravity anomalies

Wojciech Jarmołowski
  • Corresponding author
  • University of Warmia and Mazury Faculty of Geodesy and Land Management Department of Satellite Geodesy and Navigation ul. Heweliusza 5, 10-724 Olsztyn, Poland
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Published Online: 2013-12-31 | DOI: https://doi.org/10.2478/geocart-2013-0013


The paper describes the estimation of covariance parameters in least squares collocation (LSC) by the cross-validation (CV) technique called leave-one-out (LOO). Two parameters of Gauss-Markov third order model (GM3) are estimated together with a priori noise standard deviation, which contributes significantly to the covariance matrix composed of the signal and noise. Numerical tests are performed using large set of Bouguer gravity anomalies located in the central part of the U.S. Around 103 000 gravity stations are available in the selected area. This dataset, together with regular grids generated from EGM2008 geopotential model, give an opportunity to work with various spatial resolutions of the data and heterogeneous variances of the signal and noise. This plays a crucial role in the numerical investigations, because the spatial resolution of the gravity data determines the number of gravity details that we may observe and model. This establishes a relation between the spatial resolution of the data and the resolution of the gravity field model. This relation is inspected in the article and compared to the regularization problem occurring frequently in data modeling.


Artykuł opisuje estymację parametrów kowariancji w kolokacji najmniejszych kwadratów (LSC) przy pomocy techniki kroswalidacji nazywanej leave-one-out (LOO). Wyznaczane są dwa parametry modelu Gaussa-Markova trzeciego rzędu (GM3) wraz z odchyleniem standardowym szumu a priori, które ma znaczny wpływ na macierz kowariancji złożoną z sygnału i szumu. Testy numeryczne przeprowadzono na dużym zbiorze anomalii grawimetrycznych Bouguera z obszaru centralnej części USA. Obszar ten mieści około 103000 pomiarów grawimetrycznych. Dane te wraz z regularnymi siatkami wygenerowanymi z modelu geopotencjalnego EGM2008 pozwalają na pracę z różną rozdzielczością przestrzenną i różnymi wariancjami sygnału i szumu. Odgrywa to kluczową rolę w badaniach numerycznych, ponieważ rozdzielczość przestrzenna danych grawimetrycznych wyznacza liczbę szczegółów pola siły ciężkości, które możemy obserwować i modelować. Oznacza to relację pomiędzy rozdzielczością przestrzenną danych i rozdzielczością modelu pola siły ciężkości. Związek ten jest w artykule analizowany i porównywany z problemem regularyzacji, występującym często w modelowaniu danych przestrzennych.

Keywords : gravity anomaly; least squares collocation; leave-one-out; covariance; noise

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About the article

Published Online: 2013-12-31

Published in Print: 2013-12-01

Citation Information: Geodesy and Cartography, ISSN (Print) 2080-6736, DOI: https://doi.org/10.2478/geocart-2013-0013.

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