Is it really a Mogi or a mirage caused by the assumption of medium homogeneity?

Surface deformation in volcanic areas is often ascribed to inflation/deflation of pressurized cavities. Fast data inversion requires running many forward models based on the approximate computation of surface deformation due to selected expansive sources (sphere, spheroid, sill, and, more recently, scalene ellipsoid) embedded in a homogeneous elastic half-space. A simple very small spherical source (Mogi’s model; Mogi, 1958) often satisfy observations reasonably.

However, Earth is not homogeneous but characterized by vertical and horizontal heterogeneities; in many cases, vertical heterogeneities dominate and rigidity of the medium increases with depth. On the basis of these considerations, to highlight how the assumption of homogeneous elastic medium affects the inverted source parameters we (i) calculate the surface deformation due to point spheroids with vertical polar axis embedded in a layered half-space, and (ii) invert the computed surface deformations for the same source types (but with free geometry) embedded in a homogeneous half-space. In both forward and inverse modeling, the point source is schematized using an appropriate moment tensor (Davis, 1986); in forward modelling, we use Wang's Green functions (Wang et al., 2006) and, for comparison, also FEM. We consider two different examples of layering, which are approximately valid for Campi Flegrei (Amoruso et al., 2008) and Long Valley (velocity profiles from Biondi et al., 2023) respectively, and invert computed surface deformation (radial and vertical displacements) up to various distances from the source axis. We minimize both the mean square deviation (L2 norm) and the mean absolute deviation (L1 norm) of residuals.

The inversions show that:

1. as expected, the retrieved source appears shallower than the “real” source (i. e., focussing effect); the focussing effect depends on the source aspect ratio, generally increasing from prolate to oblate spheroids;

2. unless the “real” source is strongly vertically elongated (i. e. prolate), the retrieved source always appears very close to a spherical one (Mogi’s model); this effect depends on the source depth (less strong for shallow sources) and inversion norm (less strong for L1 norm).

Our results explain why a Mogi’s source so often satisfy deformation data, but also rise a big warning on the reliability of the inversions that provide it as a solution.

References

Amoruso, A., Crescentini, L., Berrino, G. (2008), Earth Planet. Sci. Lett., 272, 181-188.

Biondi, E. et al. (2023), Sci. Adv. 9, eadi9878.

Davis, P. M. (1986), J. Geophys. Res., 91, 7429-7438.

Mogi, K. (1958), Bull. Earthq. Res. Inst. 36, 99-134.

Wang, R., Lorenzo Martín, F., Roth, F. (2006), Comput. Geosci. 32, 527-541.