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An equipotential surface of Earth gravity, including the centrifugal potential, used as an idealized sea-level reference. geoid height responds to long-wavelength density anomalies and boundary deformation; local gravity and geoid height are related but distinct observables.

Homologous temperature and deformation. Homologous temperature is $\Theta =T/T_m$, with both temperatures in kelvin and $T_m$ the appropriate pressure- and composition-dependent melting temperature. It measures proximity to melting more meaningfully than an absolute temperature alone. Pressure, composition and water can alter both $T_m$ and the flow law, so equal $\Theta$ does not imply identical rheology.

A useful thermally activated material creep law iswith stress $\sigma$, grain size $d$, activation energy $E^{\ast }$ and activation volume $V^{\ast }$. Water activity can add another factor. Increasing temperature raises the rate at fixed stress when the activation enthalpy is positive, greatly lowering effective viscosity. Solid material can therefore flow on geological timescales well below melting.

Diffusion creep is approximately linear in stress: bulk-diffusion Nabarro-Herring creep has grain-size scaling $d^{-2}$, while grain-boundary Coble creep has $d^{-3}$ in their ideal regimes. Dislocation creep usually has a stress exponent greater than one and weaker grain-size dependence. Its effective viscosity scales as ${\eta }_{\mathrm{eff}}\propto {\sigma }^{1-n}$ at fixed temperature, so stronger forcing changes the apparent viscosity. At lower homologous temperature, brittle/frictional deformation or high-stress dislocation mechanisms can become important. The dominant process also depends on pressure and confinement; low temperature is not by itself a guarantee of brittle failure deep in a planet.

A deformation mechanism map compares the competing rates or strengths as functions of stress and homologous temperature at specified grain size, pressure and composition. Fine grains favour diffusion creep; larger stress favours nonlinear dislocation creep. There is no universal temperature dividing all these regimes. This sensitivity helps explain a strong outer lithosphere above a slowly flowing solid mantle and why modest thermal or water variations can reorganize deformation.

Determining the actual planform of mantle convection. The task is to infer the geometry of upwellings, downwellings and connecting flow in the present Earth, rather than to draw a preferred laboratory convection cell. Plate velocities and the distribution of subduction zones directly constrain surface motion and likely cold downwellings. Ridges, volcanism and hotspots provide further clues, but a ridge need not mark the centre of a deep plume, nor is every hotspot uniquely explained by one plume geometry.

Seismic tomography supplies three-dimensional velocity structure from body-wave travel times, waveforms and surface-wave dispersion. Fast slab-like structures can often be traced beneath subduction zones into the lower mantle. The 1997 global P-wave study reported long narrow lower-mantle structures connected with convergence and interpreted some slab segments as reaching the deepest mantle; that is evidence for deep circulation, not proof that every slab follows one route.^[ref] Temperature affects seismic speed, but composition, phase changes, water and anisotropy also matter, so velocity is not itself a flow map.

Gravity and the geoid constrain the spatially integrated density field and its induced boundary deformation. Dynamic topography supplies information about vertical stresses, after separating crustal structure, isostatic loads and thermal subsidence. Seismic anisotropy can constrain deformation and fabric orientations, subject to a mineral-physics relation between fabric and flow. Discontinuity topography and the mantle's heat budget provide further tests. Each observation has a different sensitivity to density, temperature and viscosity, making their combination more useful than any one image.

Compare these observations with three-dimensional geodynamic models that solve creeping-flow force balance, mass conservation and thermal transport, including buoyancy from temperature and composition, realistic plate motions, viscosity structure and phase boundaries. Predict seismic observables, gravity, geoid and dynamic topography from the same model, and assess data coverage and resolution before interpreting a feature. Whole-mantle exchange, impeded or deflected slabs near transition zones, and broad deep anomalies can coexist; the evidence does not require identical stacked or single-layer cells everywhere. The actual planform is constrained by joint seismic, gravitational, surface-kinematic and thermal evidence, with substantial nonuniqueness in its detailed flow and history.

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.