Deformation from Inflation of a Dipping Finite Prolate Spheroid in an Elastic Half‐Space as a Model for Volcanic Stressing

TLDR: In this article, an analytic expression for the deformation field resulting from the inflation of a finite prolate spheroidal cavity in an infinite elastic medium is given, which is equivalent to that generated by a parabolic distribution of double forces and centers of dilatation along the sphroid generator.

Abstract: Exact analytic expressions are given for the deformation field resulting from inflation of a finite prolate spheroidal cavity in an infinite elastic medium. The field is equivalent to that generated by a parabolic distribution of double forces and centers of dilatation along the spheroid generator. Approximate, but quite accurate, solutions for a dipping spheroid in an elastic half-space are found using the half-space double force and center of dilatation solutions. We compare results of the surface deformation field with those generated by the point source ellipsoidal model of Davis (1986). In the far field both models give identical results. In the near field the finite model must be used to calculate displacements and stresses within the medium. We also test the limits of applicability of the finite model as it approaches the surface by comparing the surface displacement field from a vertical spheroid with that calculated from the finite element method. We find the model gives a satisfactory approximation to the finite element results when the minimum radius of curvature of the upper surface is less than or equal to its depth beneath the free surface. Comparison of surface displacements generated by the point and finite element models gives good agreement, provided this criterion is satisfied. We have used the finite model to invert deformation data from Kilauea volcano, Hawaii. The results, which compare favorably with those obtained from the point ellipsoid model, can be used to estimate the distribution of stresses within the volcano in the near field of the source.