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

TL;DR

This paper develops a rigorous mathematical framework for modeling surface deformations caused by magma chambers in Earth's crust, generalizing the Mogi model to arbitrary cavity shapes and providing asymptotic analysis.

Contribution

It establishes well-posedness, integral solutions, and asymptotic formulas for surface deformation due to embedded cavities of arbitrary shape, extending the classical Mogi model.

Findings

Derived the principal asymptotic term for surface deformation as cavity size shrinks.

Generalized the Mogi point source model to arbitrary cavity shapes.

Provided a rigorous mathematical proof of the model's validity.

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 $\bm{z}$ and has diameter $\varepsilon>0$, small with respect to the depth $d$, we derive rigorously the principal term in the asymptotic expansion for the surface deformation as $\varepsilon=r/d\to 0^+$. 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.