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 (
,
source radius
source depth) the model predicts a purely radial displacement at the surface:

(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 [KirchdÃ¶rfer 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 *x*_{s} and the depth *z*_{s} of the source are

(21) |

where

2001-09-21