Open AccessChoice of norm for the density distribution of the Earth
Author(s) -
Sansò F.,
Barzaghi R.,
Tscherning C. C.
Publication year - 1986
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1986.tb04550.x
Subject(s) - mathematics , spectral density , norm (philosophy) , mathematical analysis , inverse , covariance , inverse problem , probability density function , gravitational field , physics , statistics , geometry , classical mechanics , political science , law
Summary. The determination of the density distribution of the Earth from gravity data is called the inverse gravimetric problem. A unique solution to this problem may be obtained by introducing a priori data concerning the covariance of density anomalies. This is equivalent to requiring the density to fulfil a minimum norm condition. The generally used norm is the one equal to the integral of the square of the density distribution ( L 2 ‐norm), the use of which implies that blocks of constant density are uncorrelated. It is shown that for harmonic anomalous density distributions this leads to an external gravity field with a power spectrum (degree‐variances) which tends too slowly to zero, i.e. implying gravity anomalies much less correlated than actually observed. It is proposed to use a stronger norm, equal to the integral of the square sum of the derivatives of the density distribution. As a consequence of this, base functions which are constant within blocks, are no longer a natural choice when solving the inverse gravimetric problem. Instead a block with a linearly varying density may be used. A formula for the potential of such a block is derived.