Models of ground deformation from vertical volcanic conduits with application to eruptions of Mount St. Helens and Mount Etna

We present simple analytical models of ground deformation from inflation of vertical volcanic conduits or pipes. We compare three cases corresponding to (1) a pressurized pipe with closed end at the top that resists the internal pressure, (2) a pressurized pipe with top open for which the internal deformation is mainly dislocation of the cylindrical walls, and (3) a pipe‐shaped region that dilates. For the closed pipe we use Eshelby's inclusion theory to model deformation from a thin, pressurized, elongated, prolate ellipsoid in a full‐space, which we express in terms of double forces. To satisfy surface boundary conditions, we superimpose image solutions by using double forces derived from Mindlin's half‐space solution for the point force. For the open pipe we apply the Volterra integral to dislocation across a cylindrical surface and generalize it to the half‐space using Mindlin's point force solution. These two solutions show marked differences from the line of dilatation solution. Ratios of maximum horizontal deformation to maximum vertical deformation for the pipe models are significantly greater than values for the center or line of dilatation models. This, and the more gradual fall off of the deformation with distance may be used as diagnostics for discriminating between pipe‐like sources and dilatational sources on volcanoes, where both components of the deformation field are available. As examples, we compare the closed pipe model with deformation associated with dome building on Mount St. Helens volcano, and tilt predicted by the open pipe model with tilt measured during a period of explosive eruptive activity on Mount Etna, Sicily, in 1995. Comparison between model values and the measurements suggests that the effective elastic moduli of the volcanic cones are very low.