Elastic stress and deformation near a finite spherical magma body: Resolution of the point source paradox

Abstract

Approximate solutions are obtained for the stress and displacement fields due to a pressurized spherical cavity in an elastic half‐space. The solutions take the form of series expansions in powers of ε = a/d, where a is the cavity radius and d is the depth. The leading‐order term in the expression for the surface uplift, which arises at O(ε 3), recovers the well‐known result of Mogi for the response to a point dilatation. The first higher‐order correction accounts for a cavity of finite size and thus offers the possibility of fitting leveling data for not only the depth but also the radius and pressure increment. However, this correction is of O(ε 6) and, consequently, is weak. The result provides a formal explanation for the success of the point dilatation model in representing uplift data even when it is known independently that ε is not small. The higher‐order correction causes the surface uplift to fall off more rapidly in the radial direction, implying that a fit of the point source solution tends to underestimate the depth d. In contrast to the surface displacement, the stress field near the cavity is affected profoundly by the proximity of the free surface. Three higher‐order corrections to the stress field are obtained, which result in a uniformly valid approximation to O(ε 5). The hoop stress at the cavity exhibits a tensile maximum at the circle of tangency with a cone with its apex at the free surface. This result appears to be consistent with the locus of fractures radiating outward from the magma body inferred by seismic methods in Long Valley, California.