Index theory-based algorithm for the gradiometer inverse problem
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Index theory-based algorithm for the gradiometer inverse problem Robert C. Anderson1 · Jonathan W. Fitton1
Received: 29 December 2014 / Accepted: 6 March 2015 © Springer-Verlag Berlin Heidelberg (outside the USA) 2015
Abstract We present an Index Theory based gravity gradiometer inverse problem algorithm. This algorithm relates changes in the index value computed on a closed curve containing a line field generated by the positive eigenvector of the gradiometer tensor to the closeness of fit of the proposed inverse solution to the mass and center of mass of the unknown. We then derive a method of determining bounds on the unknown’s center of mass and/or total mass and apply it as a function of gradiometer observables. Both observational errors and the varieties of possible mass distributions generating the gradients are taken into account for the bounds. Keywords
Gravity gradiometer · Inverse problem · Index Theory
Mathematics Subject Classification
31A99
1 Introduction There has been a surge of interest in application involving gradiometer data recently, particularly gradiometer inverse problems. One main areas of application surrounds gradiometer inverse problems focused on the European Space Agency’s geodetic satellite mission, GOCE, and the gravitational gradient observations produced along its orbits. This inverse problem is interested in producing spherical harmonic series rep-
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Robert C. Anderson [email protected] Jonathan W. Fitton [email protected]
1
National Geospatial Intelligence Agency, Basic and Applied Research Office, NGA/IB, 3838 Vogel Rd, MS L-64, Arnold, MO 63129, USA
123
Int J Geomath
resentations of the Earth’s gravitational potential (Freeden and Nutz 2011; Murböck et al. 2011; Novák and Tenzer 2013). The second main focus area is on geophysical prospecting problems now feasible due to recent improvements in airborne and land gradiometers, such as those available from Lockheed Martin (Difrancesco 2007). Additionally, emerging systems based on atom interferometry show promise at increasing instrument accuracy by an order of magnitude (McGuirk et al. 2002). Also advances in algorithms aimed at modeling the gradients from local terrain and improving the likelihood of solving the local prospecting inverse problem are presented in Jekeli (2012) and Uzun and Jekeli (2015). The inverse source problem for the gradiometer tensor can be stated generally as follows: given a gradiometer tensor field, extract information about the unknown object from which it was generated. In practice, information about the unknown object is determined by identifying model parameters that generate gradiometer terms at the surveyed locations that are a close fit to the observations. This may be done by some estimation process like least squares or by trying many models (forward modeling) and keeping a “best fitting” one. This model is then assumed to have something in common with the unknown object—location, mass, etc. In Anderson (2011), it was shown that the gradiometer inverse problem reduc
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