Analysis of a Mogi-type model describing surface deformations induced by a magma chamber embedded in an elastic half-space

Abstract

Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in an half-space with an embedded cavity. The boundary conditions are traction-free for the air/crust boundary and uniformly hydrostatic for the crust/chamber boundary. These are complemented with zero-displacement condition at infinity (with decay rate).After a short presentation of the model and of its geophysical interest, we establish the well-posedness of the problem and provide an appropriate integral formulation for its solution for cavity with general shape. Based on that, assuming that the chamber is centred at some fixed point z and has diameter r > 0, small with respect to the depth d, we derive rigorously the principal term in the asymptotic expansion for the surface deformation as ε = r/d → 0 + . Such formula provides a rigorous proof of the Mogi point source model in the case of spherical cavities generalizing it to the case of cavities of arbitrary shape.

Analysis of a Model of Elastic Dislocations in Geophysics

“…Following the same calculations as, for example, in [8,Theorem 3.3] and employing the local estimates (81), it is straightforward to prove that (82)…”

“…We denote the Neumann function for a homogeneous and isotropic half space by N 0 (x, y). Its explicit expression can be found, for instance, in [30,31,8,27]. Moreover, we assume S is a rectangle parallel to the plane {x 3 = 0}, that is,…”

On an elastic model arising from volcanology: An analysis of the direct and inverse problem

“…Before proving this theorem, we briefly recall the integral representation formula for the solution to problem (4) derived in [7]. In particular, we define first the Neumann function, solution to ∇N(·, y) ∇N(·, y))n = 0 on R 2 N = O(|x| −1 ),…”

“…Proposition 4.6 (Stability Estimates of Continuation from Cauchy Data). Let C 1 and C 2 be two domains satisfying (7), (8), (9) and (34). Moreover, let w i , for i = 1, 2, be the solution to (37) with C = C i .…”

“…In this paper we investigate a linear elastic model describing surface deformations in a volcanic area induced by a magma chamber embedded in the Earth's crust. From the mathematical point of view we introduce a simplified version of the model assuming the crust to be a half-space in a homogeneous and isotropic medium and the magma chamber by a cavity subjected to a constant pressure on its boundary (for more details see, for example, [7,8,12,17]). More precisely, given C a fourth-order isotropic and homogeneous elastic tensor, with Lamé parameters λ and µ, denoting by u the displacement vector and R 3 − the half-space, we end up with the following linear elastostatic boundary value problem…”