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The problem

Gerhard Müller (2001) has noted an apparent inconsistency between two relationships used to calculate the volume change $\Delta V$of a seismic source from the isotropic part of its moment-tensor representation. The diagonal elements M of the isotropic tensor Mi are one-third of the trace of the original tensor M=(mij) each.

\begin{displaymath}\mathbf{M_i}=\left( \begin{array}{ccc}
M & 0 & 0 \\ 0& M & 0...
...M
\end{array} \right)
, \qquad M = \frac{1}{3} \sum m_{jj}
\end{displaymath} (1)

The relationships in question are

\begin{displaymath}M = (\lambda + 2 \mu / 3) \ \Delta V
\end{displaymath} (2)

following Bowers and Hudson (1999) and others (for references see Müller 2001), and

\begin{displaymath}M = (\lambda + 2 \mu) \ \Delta V
\end{displaymath} (3)

following Müller (1973) and equivalent formulae for the seismic far-field de­rived by Wielandt (1972, 1975). Both formulae have been applied to spherical sources; however for $\mu > 0$ eq. (2) gives a larger source volume than eq. (3). The apparent inconsistency has not fully been resolved in Müller's note; he shows however that (2) applies to the moment-tensor equivalent of a crack and (3) to the moment-tensor equivalent of a spherical explosive source.


next up previous contents
Next: Transformational and distributed volume Up: Draft On the relationship Previous: Contents
Erhard Wielandt
2003-05-30