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\b[<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 $2\pi G\rho_ch$. <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 $\rho_ch=(\rho_m-\rho_c)b$. 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 approximately
$$
Z_{\rm FA}(k)=2\pi G\rho_c(1-e^{-kD}),\qquad Z_{\rm B}(k)=-2\pi G\rho_ce^{-kD}.
$$
Long 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 $q$, $D_f\nabla^4w+\Delta\rho g w=q$, with $D_f=ET_e^3/[12(1-\nu^2)]$. Thus $w_k=q_k/(D_fk^4+\Delta\rho g)$, 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.<The relationship between bathymetry and gravity in the Atlantic Ocean by Dan McKenzie and Carl Bowin>[pp. 1903–1915, ship-track sampling, spectral transfer estimates and wavelength-dependent compensation]{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.
\b[<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 $\tau_k\sim2\eta k/(\rho g)$. 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.<Space geodesy constrains ice age terminal deglaciation: The global ICE-6G_C (VM5a) model by W. R. Peltier and others>[abstract and pp. 450–487, joint ice-history/GIA model and space-geodetic constraints]{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. \b[Combine spatially and temporally distinct observations; there is no single directly measured <mantle viscosity> independent of depth, <rheology> and loading assumptions.]
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