Shape and volume change of pressurized ellipsoidal cavities from deformation and seismic data

(W)e present exact expressions for the volume change of a pressurized ellipsoidal cavity in an infinite homogeneous elastic medium. The expressions can be used as approximate solutions also for a homogeneous half-space. We show that previously published widely used expressions are correct for spheres but underestimate the ratio of the volume change to the product of pressure and volume in any other case. We discuss the capability to infer the shape of a single ellipsoidal cavity from far-field deformation measurements. Our results indicate that source axis ratios may often be hard to estimate, whereas it may be easier to infer the volume change of the source. We also consider the case of a source region consisting of interconnected pressurized ellipsoidal cavities, neglecting mutually induced stress. If all the cavities share the same shape and orientation, the source is seen in the far field as a single ellipsoidal cavity and it is possible to compute the total volume change from surface displacements. The shape of the apparent single ellipsoid is the same as the shape of the constituting cavities and not of the source region. In any other case a single ellipsoidal cavity might even be unable to give the same surface displacements as the ensemble of cavities. Since sudden volume change of a cavity can generate seismic waves, we discuss the decomposition of the related moment tensor into isotropic, double-couple, and compensated linear-vector dipole force systems in case of magma exchange between two ellipsoidal cavities, giving relations for the moment tensor components.