A parameter choice strategy for the inversion of multiple observations
Article
First Online:
Received:
Accepted:
- 108 Downloads
Abstract
In many geoscientific applications, multiple noisy observations of different origin need to be combined to improve the reconstruction of a common underlying quantity. This naturally leads to multi-parameter models for which adequate strategies are required to choose a set of ‘good’ parameters. In this study, we present a fairly general method for choosing such a set of parameters, provided that discrete direct, but maybe noisy, measurements of the underlying quantity are included in the observation data, and the inner product of the reconstruction space can be accurately estimated by the inner product of the discretization space. Then the proposed parameter choice method gives an accuracy that only by an absolute constant multiplier differs from the noise level and the accuracy of the best approximant in the reconstruction and in the discretization spaces.
Keywords
Parameter choice Multiple observations Spherical approximationPreview
Unable to display preview. Download preview PDF.
References
- 1.Bauer, F., Gutting, M., Lukas, M.A.: Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics. 2nd edn. Springer (2015)Google Scholar
- 2.Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problems. Math. Comp. 81, 1795–1841 (2011)MathSciNetMATHGoogle Scholar
- 3.Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Num. Math. 94, 203–228 (2003)MathSciNetCrossRefMATHGoogle Scholar
- 4.
- 5.Chen, J., Pereverzyev, S. Jr., Xu, Y.: Aggregation of regularized solutions from multiple observation models. Inverse Prob. 31, 075005 (2015)MathSciNetCrossRefMATHGoogle Scholar
- 6.Drinkwater, M.R., Floberghagen, R., Haagmans, R., Muzi, D., Popescu, A.: GOCE: ESA’s first Earth explorer core mission. In: Beutler, G.B., Drinkwater, M.R., Rummel, R., Von Steiger, R. (eds.) Earth Gravity Field from Space - from Sensors to Earth Sciences. Kluwer Academic Publishers (2003)Google Scholar
- 7.Driscoll, J.R., Healy, M. H. Jr.: Computing fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15, 202–250 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 8.Freeden, W.: On approximation by harmonic splines. Manuscr. Geod. 6, 193–244 (1981)MATHGoogle Scholar
- 9.Freeden, W.: Multiscale modelling of spaceborne geodata (1999)Google Scholar
- 10.Friis-Christensen, E., Lühr, H., Hulot, G.: Swarm: A constellation to study the Earth’s magnetic field. Earth Planet. Space 58, 351–358 (2006)CrossRefGoogle Scholar
- 11.Gerhards, C.: A combination of downward continuation and local approximation for harmonic potentials. Inverse Prob. 30, 085004 (2014)MathSciNetCrossRefMATHGoogle Scholar
- 12.
- 13.Haines, G.V.: Spherical cap harmonic analysis. J. Geophys. Res. 90, 2583–2591 (1985)CrossRefGoogle Scholar
- 14.Hesse, K., Sloan, I., Womersley, R.S.: Numerical integration on the sphere. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics. Springer (2010)Google Scholar
- 15.Hesse, K., Womersley, R.S.: Numerical integration with polynomial exactness over a spherical cap. Adv. Comp. Math. 36, 451–483 (2012)MathSciNetCrossRefMATHGoogle Scholar
- 16.Koch, K.R., Kusche, J.: Regularization of geopotential determination from satellite data by variance components. J. Geod. 76, 259–268 (2002)CrossRefMATHGoogle Scholar
- 17.Lu, S., Pereverzyev, S.: Multi-parameter regularization and its numerical realization. Num. Math. 118, 1–31 (2011)MathSciNetCrossRefGoogle Scholar
- 18.Lu, S., Pereverzyev, S.: Multiparameter regularization in downward continuation of satellite data. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics. 2nd edn. Springer (2015)Google Scholar
- 19.Merrick, J.R.: Aggregation of forecasts from multiple simulation models. In: Pasupathy, R., Kim, S.-H., Tolk, A., Hill, R., Kuhl, M.E. (eds.) Proceedings of the 2013 Winter Simulation Conference (2013)Google Scholar
- 20.Pavlis, N.K., Holmes, S.A., Kenyon, S.C., Factor, J.K.: The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res. 117, B04406 (2012)Google Scholar
- 21.Pereverzyev, S., Schock, E.: Error estimates for band-limited spherical regularization wavelets in an inverse problem of satellite geodesy. Inverse Probl. 15, 881–890 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 22.Pilkington, M.: Aeromagnetic surveying. In: Gubbins, D., Herrero-Bervera, E. (eds.) Encyclopedia of Geomagnetism and Paleomagnetism. Springer (2007)Google Scholar
- 23.Shure, L., Parker, R.L., Backus, G.E.: Harmonic splines for geomagnetic modeling. Phys. Earth Planet. Inter. 28, 215–229 (1982)CrossRefGoogle Scholar
- 24.Simons, F.J., Dahlen, F.A., Wieczorek, M.A.: Spatiospectral localization on a sphere. SIAM Rev. 48, 505–536 (2006)CrossRefMATHGoogle Scholar
- 25.Thébault, E., Schott, J.J., Mandea, M.: Revised spherical cap harmonic analysis (RSCHA): validation and properties. J. Geophys. Res. 111, B01102 (2006)Google Scholar
Copyright information
© Springer Science+Business Media New York 2016