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

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; Rösler, R., 1981. Über die Fehlerfortpflanzung bei Potentialfeldtransformationen. Gerlands Beitr. Gephys. 90, 47-57).