Deformation of solid earth by surface pressure: equivalence between Ben-Menahem and Singh’s formula and Sorrells’ formula

Toshiro Tanimoto

Published June 6, 2024, SCEC Contribution #14156

Atmospheric pressure changes on Earth’s surface can deform the solid Earth. Sorrells derived analytical formulae for displacement in a homogeneous, elastic half-space, generated by a moving surface pressure source with speed
c. Ben-Menahem and Singh derived formulae when an atmospheric P wave impinges on Earth’s surface. For a P wave with an incident angle close to the grazing angle, which essentially meant a slow apparent velocity ca in comparison to P-wave ( α  ) and S-wave velocities ( β  ) in the Earth ( ca << β  < α  ), they showed that their formulae for solid-Earth deformations become identical with Sorrells’ formulae if ca is replaced by c. But this agreement was only for the asymptotic cases ( ca << β ). The first point of this paper is that the agreement of the two solutions extends to non-asymptotic cases, or when ca /β  is not small. The second point is that the angle of incidence in Ben-Menahem and Singh’s problem does not have to be the grazing angle. As long as the incident angle exceeds the critical angle of refraction from the P wave in the atmosphere to the S wave in the solid Earth, the formulae for Ben-Menahem and Singh’s solution become identical to Sorrell’s formulae. The third point is that this solution has two different domains depending on the speed c(or ca) on the surface. When c/β  is small, deformations consist of the evanescent waves. When c approaches Rayleigh-wave phase velocity, the driven oscillation in the solid Earth turns into a free oscillation due to resonance and dominates the wave field. The non-asymptotic analytical solutions may be useful for the initial modelling of seismic deformations by fast-moving sources, such as those generated by shock waves from meteoroids and volcanic eruptions because the condition c/β << 1 may be violated for such fast-moving sources.

Citation
Tanimoto, T. (2024). Deformation of solid earth by surface pressure: equivalence between Ben-Menahem and Singh’s formula and Sorrells’ formula. Geophysical Journal International, 238(2), 820-826. https://doi.org/10.1093/gji/ggae185.