Gravity-topography admittance ($Z(k)$)

A spectral transfer estimate $Z(k)=⟨g_k{h}_k^{\ast }⟩/⟨|h_k{|}^2⟩$. Its wavelength dependence constrains compensation, density structure and support, but one transfer curve does not uniquely determine them.

For cross-spectrum $S_{gh}$ and autospectra $S_{gg},S_{hh}$, squared coherence is $|S_{gh}{|}^2/(S_{gg}{S}_{hh})$. It measures the fraction of spectral association between gravity and topography. Low coherence can signal uncorrelated internal loading, noise or shortcomings of a simple surface-load model; interpreting it as elastic thickness requires a loading model.

For a surface load above a uniform-density crust over a denser substrate, a simple lithospheric flexure model gives the indicated gravity-topography admittance, with ${\ell }_f^4=D/[g({\rho }_m-{\rho }_c)]$. The surface mass contributes positive gravity; downward displacement of the density interface contributes a compensating negative signal attenuated by $e^{-kt}$. Hence $Z\to 0$ at long wavelength and $Z\to 2\pi G{\rho }_c$ at short wavelength. Here $t$ is the interface depth, not automatically the effective elastic thickness.