On the differential properties of internal magnetic field models at the Earth's surface and at satellite altitudes

摘要

The inverse theory of potential fields shows that the correspondence between the internal magnetic field of the Earth and its field sources is unique when the potential field is known in all points of the three-dimensional space including all points of the source region (cp., e.g. Diesselhorst, H., 1939. Magnetische Felder und Kraefte. Johann Ambrosius Barth Verlag, Leipzig). Thus, to determine the sources of the field it is not sufficient to know the potential field in the space external to the sources. Moreover, field models derived from finite sets of potential field observations emphasize different source properties because of measurement errors. In this study, I argue that improved internal field models can be developed from multi-altitude magnetic observations by imposing more effective constraints on the poorly conditioned downward continuation problem. In particular, the convergence behaviour of spherical harmonic field models can be used to improve the downward continuation of the higher truncation index terms. A high quality approximation of the field continuation is essential when the field models are interpreted for relatively small field contributions such as from the lithospheric sources. The relations between the potential field and its sources including the problems of potential field continuations - upward and downward - are governed by the theory of ill-posed inverse problems (cp., e.g. Anger, G., 1990. Inverse Problems in Differential Equations. Akademie/Plenum Press, Berlin/London; Anger, G., Gorenflo, R., Jochmann, H., Moritz, H., Webers, W. (Eds.), 1993. Inverse Problems: Principles and Applications in Geophysics, Technology, and Medicine. Akademie, Berlin; Huestis, S.P., Parker, R.L., 1979. Upward and downward continuations as inverse problems. Geophys. J. R. Astr. Soc. 57, 171-188; Roesier, R., 1981. Ueber die Fehlerfortpflanzung bei Potentialfeldtransformationen. Gerlands Beitr. Gephys. 90, 47-57).