In this talk I will present a linear elastic model describing ground’s deformations within calderas due to an inflating magma chamber. This analytical model replaces caldera with a homogeneous, isotropic half-space and the magma chamber by a pressurized cavity. In the first part of the talk I will exhibit results on the well-posedness of this problem within the framework of layer potentials techniques and weighted Sobolev spaces. Then, using the physical hypothesis for which the pressurized cavity is small with respect to its depth, the asymptotic analysis of the solution of the problem will be showed. In the second part, considering a single measurement of the displacement taken on a portion of the boundary of the half-space, the inverse problem of determining uniquely and in a stable way the pressurized cavity will be addressed. Finally, I will give some details on the first attempts of implementing a reconstruction algorithm which can identify the pressurized cavity.