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For finite mantle depth, retain all four vertical solutions of the biharmonic equation,rather than discarding the terms that grow as $z\to -\infty$. Apply the plate conditions at $z=0$ and matching conditions at $z=-H$. At the mantle--core interface these include continuity of normal velocity and normal stress, an appropriate tangential-stress condition for the liquid core, and a kinematic boundary condition for interface displacement. The normal-stress balance gains the stable buoyancy termbecause ${\rho }_c>\rho$.

The resulting homogeneous linear system for the mode amplitudes has a nontrivial solution only when its determinant vanishes; that condition replaces the half-space decay rate. Modes with $kH\gg 1$ decay before sensing the core and recover the half-space result. Modes with $kH\lesssim 1$ involve the full mantle depth, feel the basal density contrast and core mobility, and acquire a modified relaxation time. This is the planar finite-depth extension of flexural isostasy.

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.

Using the preceding conductive estimate, warmer incoming crust is less dense. Interpret the given $\alpha =10^{-5}{\mathrm{K}}^{-1}$ as the volumetric coefficient entering $\delta \rho /\rho =-\alpha \Delta T$. The representative contrast givesFor ${\rho }_s\simeq 3000\mathrm{kg}{\mathrm{m}}^{-3}$, that is only about $10.5\mathrm{kg}{\mathrm{m}}^{-3}$. Let $L$ be the vertical thickness of the anomalous slab material in a compensated column. Its mass deficit per unit area is approximately ${\rho }_s\alpha \Delta {\overline{T}}_{c}L$. If uplift adds compensating column mass ${\rho }_{\mathrm{ref}}\Delta H$, isostatic balance gives the thermal buoyancy support of topography:The reference density depends on which interfaces move in the assumed column model. For approximately equal densities and a near-horizontal $7$ km crustal layer, this gives $\Delta H\simeq 25$ m; using ${\rho }_s=3000$ and ${\rho }_{\mathrm{ref}}=3300\mathrm{kg}{\mathrm{m}}^{-3}$ gives about $22$ m. A modest dip changes the vertical intersection to approximately $h_c/\cos \theta$ and does not change this order of magnitude. Different graphical flux choices and uncertain geometry change the number; they do not justify replacing a $7$ km crustal anomaly by a whole $100$ km lithospheric anomaly without additional information.

Thus the crust-only thermal contrast can supply tens of metres of isostatic relief, not kilometres. Its sign would favor greater thermal support from the warmer Juan de Fuca crust, so it cannot by itself explain why much of the central Andes stands higher than Cascadia. Crustal shortening and roots, whole-slab thermal structure, phase transformations, hydration, magmatic addition and dynamic stresses can dominate. If the stated expansion coefficient instead meant a linear solid expansion coefficient, the first-order density coefficient would be about three times larger; the estimate would still be well below kilometre-scale mountain relief.

To test isostasy, jointly measure surface topography, free-air gravity anomalies, Bouguer gravity anomalies and crustal thickness from receiver functions and seismic refraction surveys. An Airy isostasy interpretation predicts crustal roots beneath high terrain and a corresponding regional negative Bouguer signal; thermal or compositional compensation predicts different density structure. Compare gravity and geoid observations with compensated-column predictions over a range of wavelengths, and use gravity-topography admittance to test local versus flexural isostasy. A small long-wavelength free-air anomaly can support compensation, but is not a sufficient test by itself. GNSS uplift, leveling and dated marine or river terraces constrain transient or sustained vertical motion; seismic tomography and rock-density estimates constrain deeper buoyancy. Residual relief or gravity after these corrections may require dynamic topography or tectonic support. An actively deforming subduction margin need not satisfy instantaneous local isostatic equilibrium, even if the estimate above is a useful scale comparison.