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# A mathematical source model

For a given isotropic moment M, we can easily construct a mathematical source model with any desired positive or negative volume change. We simply let a linear vector dipole (LVD) with the moment 3M act in a small sphere, either in its center or evenly distributed. (If the isotropic moment equals zero, we may use a suitable scaled CLVD - a compensated linear vector dipole - instead.) The displacement field outside the source sphere has the divergence

 (6)

The volume integral of this divergence is zero over any spherical shell enclosing the source, but since the divergence decays as r-3 in any given direction from the source, the integral diverges to plus or minus infinity if taken over a suitable cone in the polar or equatorial direction. Our definition of the source permits to consider any part of such a cone as part of the source. We can thus define the boundary of the source in such a way that its total volume change has any desired value.

Next: Acknowledgment Up: Draft On the relationship Previous: Source volume and moment
Erhard Wielandt
2003-05-30