DOWNWARD FIELD CONTINUATION IN COMBINING SATELLITE AND GROUND-BASED INTERNAL MAGNETIC FIELD DATA - Webers W.A. (My University, 2002)
When ground-based and satellite internal magnetic field data are available they are referred to different concentric spheres outside a sphere containing all the sources. Suppose that ground-based and satellite simultaneous data sets are treated independently. For both independent and simultaneous Spherical Harmonic Analysis (SHA) field models are calculated by the method of least squares via the set of Gauss coefficients {gnm ; hnm} where a definite truncation index N is used. Furthermore, there is supposed that the Gauss coefficients are correctly determined under these conditions so that no error discussion has to be included. By both of the models-the ground-based and the satellite SHA model-for all points outside the sphere containing all the sources the values of the internal field can be determined, but both models will give different results because for the same truncation index N there is a different approximation quality of the infinite series expansion SHA for the ground in comparison to satellite altitude. The mathematical reason is that the convergence quality of the infinite SHA expansion depends on the distance of the reference sphere from the source region to which the expansion is referred. The differences can be evaluated when the relevant functional systems for both SHA models are investigated for which the different set of Gauss coefficients are determined. If ground-based and satellite internal magnetic field data are combined it is necessary to pay attention to such differences. From the physical point of view it is clear that the physical content of the data depends on the distance of the reference sphere from the source region. Shorter wavelengths of the field are strongly attenuated at satellite altitudes. Therefore, it proves to be advantageous to include a downward field continuation when ground-based and satellite internal magnetic field data are combined although this is an ill-posed inverse problem from the mathematical point of view (Anger, G., 1990. Inverse Problems in Differential Equations. Akademie, Berlin: Plenum Press, London; Huestis, S.P. Parker, R.L., 1979. Upward and downward continuation as inverse problems. Geophys. J. R. Astr. Soc. 57, 171-188). From the convergence quality of simultaneous SHA field models referred to related concentric reference spheres a regularizing criterion for a downward continuation of the Gauss coefficients had been derived which uses inherent mathematical characteristics of the SHA procedure [Webers, W., 1998. On Combining Ground-based and Satellite Magnetic Field Data (Scientific Technical Report STR 98/21). Geo Forschungs Zentrum Potsdam (pp. 384-401); Webers, W., 1999. On determining reference fields from ground-based and satellite field data. Phys. Chem. Earth (A) 24, 943-947]. Applying this procedure to downward field continuation for the International Geomagnetic Reference Field DGRF 1990 demonstrates the different physical content of simultaneous field models at concentric spheres with their different structures. Moreover, mathematical upward and downward continuation in connection with simultaneous SHA field models based on simultaneous independent data sets at satellite altitude and at the ground will open new insights how internal and external constituents contribute to magnetic field records at different altitudes.