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The Mogi model

The greatest deficiency of the simple concepts developed so far is that they do not allow for a free surface. Yamakawa (1955) found a static solution for the displacement field of a spherical pressure source buried in an elastic halfspace. This model for volcanic sources is usually named after Mogi (1958). Under some slight simplifications ( $\lambda = \mu$, source radius $\ll$ source depth) the model predicts a purely radial displacement at the surface:

\begin{displaymath}u_r=\frac{3 a^3 P}{4 \mu r^2} = \frac{3 V}{4 \pi r^2}
\end{displaymath} (20)

It is hard to believe that the static displacement field at the free surface should differ from that in a full space (eq. 10) by nothing else than a constant factor of three. Yet the result has been checked with a finite-element computation [Kirchdörfer 1999], and there is no doubt that it is correct. Displacements and tilts measured at the free surface can therefore be interpreted as if they were observed in a full space, except that all displacements must first be divided by a factor of three.

Using the tilt as an additional observation, the essential parameters of the source (location, depth, and volume) can theoretically be inferred from broadband seismic observations at a single station on the basis of the Mogi model [Wielandt & Forbriger 1999]. The distance xs and the depth zs of the source are

\begin{displaymath}x_s=\frac{3 Z}{T} \, {\left (\frac{X}{R} \right )}^2 \, , \qquad z_s=\frac{3 X}{T} \, {\left (\frac{Z}{R} \right )}^2
\end{displaymath} (21)

where X, Z, R are the displacement amplitudes in the x, z and r directions, and T is the tilt amplitude. Practically, however, the tilt should be measured over a baseline that is longer than the scale of local hererogeneities, so a small triangle of stations is more appropriate. The horizontal displacements must be separated from the tilt signal as in Wielandt & Forbriger (1999).


next up previous contents
Next: Effects of hereogeneity and Up: Basics of the volume-source Previous: Relation with other source
Erhard Wielandt
2001-09-21