Deformation due to a pressurized horizontal circular crack in an elastic half-space, with applications to volcano geodesy

We consider deformation due to sill-like magma intrusions using a model of a horizontal circular crack in a semi-infinite elastic solid. We present ex act expressions for vertical and horizontal displacements of the free surfa ce of a half-space, and calculate surface displacements for a special case of a uniformly pressurized crack. We derive expressions for other observabl e geophysical parameters, such as the volume of a surface uplift/subsidence , and the corresponding volume change due to fluid injection/withdrawal at depth. We demonstrate that for essentially oblate (i.e. sill-like) source g eometries the volume change at the source always equals the volume of the d isplaced material at the surface of a half-space. Our solutions compare fav ourably to a number of previously published approximate models. Surface def ormation due to a 'point' crack (that is, a crack with a large depth-to-rad ius ratio) differs appreciably from that due to an isotropic point source ( 'Mogi model'). Geodetic inversions that employ only one component of deform ation (either vertical or horizontal) are unlikely to resolve the overall g eometry of subsurface deformation sources even in a simplest case of axisym metric deformation. Measurements of a complete vector displacement field at the Earth's surface may help to constrain the depth and morphology of acti ve magma reservoirs. However, our results indicate that differences in surf ace displacements due to various axisymmetric sources may be subtle. In par ticular, the sill-like and pluton-like magma chambers may give rise to diff erences in the ratio of maximum horizontal displacements to maximum vertica l displacements (a parameter that is most indicative of the source geometry ) that are less than 30 per cent. Given measurement errors in geodetic data , such differences may be hard to distinguish.