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.

The narrow volcanic ridge loads the lithosphere and generates a regional flexural response, while the broad swell involves deeper support. The observed positive ridge anomaly, negative flanks, outer positive belt, and long-wavelength positive swell anomaly probe distinct spatial scales.^[ref]

The Hawaiian swell is the broad elevation of the seafloor on which the narrower Hawaiian volcanic ridge sits. These are different spatial scales: the volcanic load bends the lithosphere through lithospheric flexure, while the broad elevation requires an explanation for deeper or more distributed support.

Ship and submarine gravity measurements show a long-wavelength positive anomaly correlated with the swell, distinct from the larger local ridge and flanking anomalies. The original analysis found that ridge crustal structure alone was unlikely to explain the long-wavelength signal and discussed compensation associated with flow beneath the lithosphere.^[ref] Combining gravity-topography admittance with other observations helps test support models; topography alone does not uniquely identify a deep mass density distribution.

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.