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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.

Oceanic thermal subsidence. Newly formed oceanic lithosphere is hot and buoyant. Cooling increases its column density, causing subsidence by isostasy as it moves away from a ridge. The half-space cooling model has profile isWith volumetric thermal expansion coefficient $\alpha$, the excess column mass relative to hot mantle isCompensation with seawater density ${\rho }_w$ givesThus young-to-intermediate-age seafloor generally deepens approximately as the square root of age. At old ages, finite thermal-plate thickness and basal heat supply flatten this trend; indefinite half-space deepening is inappropriate. Sediments, variable crust, volcanic swells and dynamic topography also perturb bathymetry. Ocean-floor depth and heat flow jointly constrain cooling models.^[ref]

Effective elastic thickness. This is the thickness of a homogeneous elastic plate with the observed effective flexural rigidity, not necessarily the crustal or thermal lithospheric thickness:Infer it by modelling deflection around known loads, such as seamounts or trenches, or fitting gravity-topography admittance and gravity-topography coherence across wavelengths. A simple Fourier balance is $(D{k}^4+\Delta \rho g)w_k=q_k$ for a signed transverse load. The transition from buoyancy-dominated to bending-dominated behaviour constrains $D$; then assumed Young's modulus $E$ and Poisson's ratio $\nu$ give $T_e$.

Good inference includes load geometry and density, buried interfaces, sediment and internal loading, boundary conditions and data resolution. These can trade off with rigidity. Plastic yielding, faulting, mechanical layering and creep make $T_e$ an effective, timescale- and stress-dependent quantity. Older, colder lithosphere is often stronger, but age at loading and subsequent relaxation matter; a single universal age-to-$T_e$ conversion is unjustified.

gravity and surface topography. gravity responds to all density structure, while surface topography records both load and support. First specify the reference field and measurement height. latitude and planetary rotation affect normal gravity. At elevation $h$, the reference field decreases approximately by $2gh/R$; a free-air gravity anomaly restores this elevation effect but retains topographic mass. A Bouguer gravity anomaly additionally subtracts an approximate topographic slab contribution $2\pi G{\rho }_{c}h$. terrain correction replaces the unrealistic slab near valleys and ridges, and spherical geometry matters over large areas. A density choice is required for these reductions; a numerical crustal density is not provided here.

In Airy isostasy, surface topography of height $h$ and crustal density ${\rho }_c$ is compensated by a root of thickness $b$ replacing mantle of density ${\rho }_m>{\rho }_c$. Equal column masses give ${\rho }_{c}h=({\rho }_m-{\rho }_c)b$. The positive gravity of the relief is partly cancelled by the negative density contrast of the root. At long wavelengths a compensated mountain can therefore have a modest free-air anomaly and a negative Bouguer anomaly. Pratt isostasy instead varies column density at a roughly common compensation depth. Neither model says that surface topography itself exerts no gravity.

Potential-field depth filtering makes wavelength crucial. In a planar Fourier approximation, a density interface at depth $z$ is attenuated by $e^{-kz}$ at horizontal wavenumber $k$. For a small Airy-compensated height perturbation and a compensation interface at depth $D$, the gravity-topography admittance is approximatelyLong wavelengths allow strong cancellation between load and root. At short wavelengths the root's gravity is filtered out, so the observed near-surface mass can dominate even if the column is compensated. In a flexural isostasy model, an elastic lithosphere supports and spreads the load; for downward deflection $w$ and imposed load $q$, $D_f{\nabla }^{4}w+\Delta \rho gw=q$, with $D_f=E{T}_e^3/[12(1-{\nu }^2)]$. Thus $w_k=q_k/(D_f{k}^4+\Delta \rho g)$, and the effective elastic thickness controls the support scale.

Estimate admittance from cross-spectra of gravity and surface topography, and compare its wavelength dependence and coherence with local-compensation, elastic-flexure and subsurface-load models. A primary Atlantic study sampled ship-track free-air gravity and bathymetry and compared their Fourier-domain relationship with compensation models, illustrating this observational route rather than inferring a root from the sign of one gravity value.^[ref] Check navigation, instrument drift, terrain/density reductions, sediment and water loads, and independent seismic crustal constraints. Density-depth trade-offs and mantle dynamic support make the inverse problem nonunique; surface topography and gravity together constrain support, but do not uniquely map every density anomaly.

mantle viscosity. A second, independent route concerns deformation rates rather than static compensation. glacial isostatic adjustment follows loading and unloading of a deformable Earth: an ice sheet depresses the surface, and subsequent removal leaves time-dependent uplift, horizontal displacement, sea-level and gravity changes. Fit dated relative sea-level indicators and raised shorelines together with geodetic displacement rates and a reconstructed ice/water load history to a viscoelastic Earth model. Relative sea level includes solid-surface motion, changing geoid and redistribution of ocean water; treating it only as local vertical rock motion biases the inference.

For scale, a sinusoidal surface perturbation of wavenumber $k$ on a uniform incompressible viscous half-space relaxes under gravity with ${\tau }_k\sim 2\eta k/(\rho g)$. A Maxwell material separately has a stress-relaxation scale ${\tau }_M=\eta /\mu$. These are different experiments and generally different times: rebound modes also involve gravity, spherical geometry, a layered mantle and elastic lithosphere. Fit a family of relaxation modes or a full load-history convolution, rather than equating every observed rebound time to $\eta /\mu$.

Use independent ice-margin, exposure-age and sea-level constraints to reduce the trade-off between ice history and viscosity; compare horizontal as well as vertical GNSS motions, and separate present ice loss, groundwater changes and tectonic signals from the older loading response. A primary global analysis jointly tested ice-history and mantle-response predictions against space-geodetic observations, rather than treating a local uplift rate as a unique viscosity measurement.^[ref]

postseismic relaxation probes shallower or faster relaxation after a stress perturbation, but must be distinguished from afterslip and poroelastic deformation. Long-wavelength geoid and mantle-flow models provide complementary constraints when density structure is estimated independently from seismic structure and mineral physics; unknown density-to-velocity conversion and boundary tractions remain trade-offs. Laboratory creep experiments constrain constitutive laws as functions of temperature, stress and grain size, but require a large extrapolation to geological conditions. Combine spatially and temporally distinct observations; there is no single directly measured mantle viscosity independent of depth, rheology and loading assumptions.