Analytical models of deformation and residual gravity changes due to a Mogi source in a viscoelastic medium

The popular Mogi model, describing deformation due to a small pressurized spherical cavity embedded in a homogeneous elastic half-space, is shortly reviewed, similarities and differences with other isotropic deformation models are discussed and the interpretation of the different volume changes accompanying source inflation is provided. The model is then generalized to viscoelastic rheology. The overpressure needed within the source to reproduce the large deformation, sometimes observed in volcanic regions, is found to be strongly dependent on rheological parameters, so that reasonably low values may be obtained if the relaxation time of the medium surrounding the source is short (months), instead of the extremely high values inferred from the elastic theory. If the inflation of the source is due to the input of new magma from remote distance, the resulting residual gravity change $Delta g$ may be easily computed, thanks to the spherical geometry of the source, and the density of the newly emplaced material may be inferred from the ratio of $Delta g/w$, between the gravity change and the uplift. This ratio is independent of the overpressure and the radius of the source, which are generally ill defined parameters, and is slightly dependent on the relaxation time in the transient phase. Explicit solutions are shown for a step-like overpressure history and for a constant magma supply rate. A more realistic model is also proposed, in which the viscoelastic properties are restricted to a small spherical volume around the source, surrounded by an elastic medium. Even if an accurate interpretation of deformation and gravity changes in volcanic areas generally requires taking into account topography, elastic layering and the triaxial geometry of the inflation source, values computed according to the generalized Mogi model are compared with observations made during the 1982-84 uplift episode at Campi Flegrei caldera (Italy), for the sake of illustration.