Surface deformation due to inflation of an arbitrarily oriented triaxial ellipsoidal cavity in an elastic half‐space, with reference to Kilauea volcano, Hawaii

Abstract

Approximate expressions for surface displacements due to fluid inflation of an arbitrarily oriented ellipsoidal cavity in an elastic half‐space are found using Eshelby's elastic inclusion theory and Mindlin's half‐space point force solution. The cavity is replaced by material having the same properties as the surrounding medium. The point force distribution on its surface is determined which exerts a uniform pressure on the medium immediately outside its boundary. Then the ellipsoid and its forces may be removed and replaced by material strained to generate the same pressure without affecting the solution in the remainder of the medium. The half‐space point force solutions were found which satisfy boundary conditions exactly on the free surface but approximately on the ellipsoid. The approximation becomes exact for a deeply buried ellipsoid and is reasonably accurate if the depth to its center is greater than twice its dimension. The far‐field solution is a weighted combination of displacements from nine double forces located at the ellipsoid center. The model was applied to measured displacement on Kilauea volcano taken over a period of intrusion and eruption, with the result that a vertically elongated ellipsoidal pressure center fits the data significantly better than a spherical one. However, earthquake swarm activity spread over the zone during the inflation episode, which demonstrates that the medium does not respond elastically. The volume of the slip found by summing seismic moments is of the same order as the surface uplift, suggesting that anelastic accommodation of the intruded material by the surrounding medium is significant. The pressure center is therefore a combination of magma pressure and failed matrix. Its vertical elongation confirms the contention that vertical elongation is expected when the volcano undergoes extreme stressing beyond the elastic limit.