C.A.Williams & G. Wadge
Environmental Systems Science Centre (ESSC), University of Reading
The analytical solution to the problem of surface deformation due to a changing pressure source buried in an elastic half space was first posed in volcanological terms by Mogi. Such models provide a first order picture of the deformation field due to magma withdrawal from a reservoir supplying an eruption and supply potentially useful limits on the depth and size of the pressurized reservoir through fitting to the observed surface deformation. In general deformation increases in magnitude as the reservoir gets shallower and decreases its effective area. Inversions of observed deformation provides strong constraints on pressure/volume change but only weak constraints on size of the reservoir.
We have taken a numerical, finite element, approach to the problem of modelling the surface deformation at Soufriere Hills Volcano. This allows us to take the finite size of the reservoir into account explicitly (as a sphere). Our models are axisymmetric and elastic. They comprise 3 horizontal layers each with different material properties to mimic the island arc crustal structure. Each finite element model has a constant reservoir pressure change applied and the elastic response calculated to include the effects of gravity and of topography. There are 3 main parameter sets to be chosen for each model: the elastic parameters of the modelling space, the pressure change based on an eruption rate from an assumed reservoir size (e.g. 2 km radius; volume = 34 km3) and the depth of the reservoir (chosen from a 3-21 km range at 3 km increments). Typical choices for the pressure change would be based on an eruption rate of about 2 x 106m3/month and the most likely depth range is between about 6 km (minimum pressure for hornblende stability) and about 16-17 km (depth to mid-crustal seismic velocity discontinuity which might be a suitable location for magma ponding).
For a model with a reservoir at 16 km depth the predicted maximum surface displacements are only of the order of a mm in magnitude for depressurizations over a period of a month. Only if the reservoir is shallower than the reasonable upper limit of 6 km should we expect surface deformations at and above the level of uncertainties (about 1 cm) of the edm and gps techniques over periods of a month or so (see first figure). The size of the deformation network on Montserrat is of limited radial extent (most stations are within a radius of 2-5 km of the magma conduit). This limits the range of displacements expected for deeper-based magma reservoirs and also any potential ability to invert the observed shape of the deformation to a particular model. This latter point, however, has remained academic because in 1996 at least this sort of regional deformation field has not been observed at Soufriere Hills Volcano. There are several possible explanations:
One possible effect that could possible account for (iv) is the surface loading due to dome growth that must complement the evacuation at depth. We have created some preliminary finite element models that explicitly include this gravitational loading effect. What we find is that the wavelength of the loading surface displacement is much shorter than the complementary deformation effect from the reservoir such that it falls to negligible levels at distances more than about 1 km from the dome (see second figure). Only the Castle Peak edm reflector is within this range. Although the shortening observed by the edm measurements from Castle Peak to Whites and Long Ground are predicted by the model the magnitude of the shortening is larger than that in the elastic loading model. The software to perform this modelling has been installed at MVO and will include the ability to calculate line length changes from any pair of measurement stations.
