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

Atmospheric pressure changes on Earth’s surface can deform the solid Earth. Sorrells derived analytical formulas for displacement in a homogeneous, elastic half-space, generated by a moving surface pressure source with speed c. Ben-Menahem and Singh derived formulas 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 formulas for solid-earth deformations become identical with Sorrells’ formulas 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 formulas for Ben-Menahem and Singh’s solution become identical to Sorrell’s formulas. 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 modeling 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.