Theory for Deriving Shallow Elasticity Structure From Colocated Seismic and Pressure Data

Colocated seismic and pressure data are available at many seismic stations in the world. For frequencies approximately between 0.01 and 0.05Hz, colocated data show evidence of strong coupling between the atmosphere and the solid Earth, especially when pressure is high. Coherence between vertical seismic data and pressure is often higher than 0.9. Such data provide information for shallow structure in the upper 50-100m because they show how the Earth responds to surface pressure changes. We present the basic theory and an inversion scheme for shallow structure using surface observables eta(f)=S-z/S-p where f is frequency and S-z and S-p are the power spectral densities of vertical seismic data and of surface pressure data. A vertically heterogeneous medium is assumed beneath a station where density, P wave velocity, and S wave velocity change with depth. We show that the integration of the minors for the equations of motion gives a fast algorithm to compute eta(f). Using numerical differentiation, we derive depth sensitivity kernels for eta(f) with which we invert eta(f) for shallow structure, just like we invert phase velocity of surface waves for the Earth structure. Depth sensitivity kernels show that if we take density, bulk modulus, and rigidity as independent parameters, eta(f) has no sensitivity to density structure, making it mainly a function of two elastic constants. We present examples of inversion based on this formulation.