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Source volume and moment

We have seen that he volume change of an opening crack is larger than the associated volume displacement through a spherical surface (section 5). For an expanding spherical source the two volumes are identical, so we do not have to distinguish between different definitions of the source volume. A source that is prolate in the direction of colinear forces has a smaller volume change than it displaces through a spherical surface. The difference is made up by the compression or expansion of the medium around the source, due to the moments acting in the source. By combining prolate and oblate sources of opposite polarity, sources may be constructed where the volume change of the source itself and the volume displacement through a spherical surface are of opposite sign.

Together with eq. (5), this implies that the isotropic part of the moment tensor may not even indicate the sign of the volume change of the source. We demonstrate this with a simple example. Consider a situation where incompressible magma is stored in a spherical chamber and then injected into a dike. The total volume of the cavities remains constant; however the isotropic moment of the chamber decreases by $ M = (\lambda + 2 \mu) \Delta V $ while the isotropic moment of the dike, idealized as a crack, increases by $ M = (\lambda + 2 \mu / 3) \Delta V $. An isotropic moment $ M = (4 \mu / 3) \Delta V $ disappears in the process; this quantity would be inferred from seismic observations as the isotropic moment of the composite source. A conventional interpretation would suggest a reduction of the magma volume, yet actually the volume reduction has occurred in the surrounding medium. The walls of the dike - which are not part of the source - were compressed while the walls of the chamber retained their volume. Alternatively, we might consider the case that the magma is compressible, and expands in the process so that the compression of the dike walls is made up. Despite this expansion of the source, the overall volume is constant and the isotropic moment is zero. Intermediate between the two cases we find expanding sources whose isotropic moment is negative. This example does not appear to be far-fetched, and should caution against an uncritical application of any moment-to-volume relationship.


next up previous contents
Next: A mathematical source model Up: Draft On the relationship Previous: The volume displacement through
Erhard Wielandt
2003-05-30