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The effective elastic thickness is the thickness of a homogeneous elastic plate with the same effective bending stiffness as the loaded lithosphere. In an isotropic thin-plate model, $D=E{T}_e^3/[12(1-{\nu }^2)]$, with Young's modulus $E$ and Poisson's ratio $\nu$. It depends on thermal state, material creep, loading history and yielding, and is not generally the crust's thickness or the full thermal thickness.

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.

Hawaii: short-wavelength volcanic loading and long-wavelength mantle support must be distinguished. A dense volcanic edifice replacing seawater adds mass and gives a positive free-air gravity anomaly. Its load bends the lithosphere down, producing a surrounding moat and an outer flexural arch. The associated displaced density interfaces contribute negative and positive gravity signals at neighbouring distances. Observed gravity near the Hawaiian ridge includes strong positive anomalies over the ridge, negative flanks, and a broader positive belt. At much longer wavelengths, the Hawaiian swell also correlates with a positive gravity anomaly, with a smaller gravity-to-topography ratio that requires compensation; the precise source of support is not determined by that correlation alone.^[ref]

The mechanical distinction can be expressed with a simple elastic-plate model. For load height $h$, downward deflection $w$, plate rigidity $D$, loading density contrast $\Delta {\rho }_l$ relative to water, and restoring density contrast $\Delta {\rho }_b$ at an interface,For a Fourier component of horizontal wavenumber $q$, this gives $w_q=\Delta {\rho }_{l}g{h}_q/(D{q}^4+\Delta {\rho }_{b}g)$. The load's direct gravity contribution competes with the negative contribution of a depressed interface at reference depth $t$. In a linear thin-interface approximation,At short wavelengths rigidity suppresses compensation, leaving relatively high positive gravity admittance. At long wavelengths the plate can deflect and compensate more completely, reducing admittance. Fitting the wavelength dependence constrains effective elastic thickness through $D=E_Y{T}_e^3/[12(1-{\nu }^2)]$, subject to density structure, load distribution and subsurface loading assumptions. This explains why local Airy compensation alone misses the spatially extended flexural response.

The broad swell needs a deeper explanation than the narrow volcanic pile: hot, buoyant mantle and mantle flow can support regional topography, while deep density anomalies and surface elevation jointly determine its gravity. A free-air anomaly keeps the attracting mass of the topography; a Bouguer correction removes an assumed direct topographic attraction and can reveal its compensating mass deficit. Gravity plus topography constrain support as a function of wavelength; neither alone uniquely fixes the mantle density distribution.

For the OR alternative, triple junctions are points where three plate boundaries meet. Each boundary can be a spreading ridge, subduction trench, or transform fault. The three plate velocities satisfyThis velocity triangle is necessary but not enough: a single junction velocity ${\mathbf{u}}_J$ must also keep the meeting point on all three moving boundaries. For symmetric spreading between A and B, the ridge constraint is $[{\mathbf{u}}_J-({\mathbf{v}}_A+{\mathbf{v}}_B)/2]\cdot \mathbf{n}=0$. For a transform, ${\mathbf{v}}_A\cdot \mathbf{n}={\mathbf{v}}_B\cdot \mathbf{n}$, and the junction must have that same normal component. For a trench, its normal migration must match the adopted subduction/rollback model; a trench fixed relative to the overriding plate shares that plate's normal velocity.

In velocity space each boundary supplies a line of possible junction velocities. A common intersection means the junction can migrate while preserving its local configuration; incompatible constraints force boundary reorganization. This is kinematic stability, not the claim that the junction is stationary. A symmetric ridge–ridge–ridge example has three arms meeting at $120^{\circ }$ and a common junction velocity; unequal spreading rates do not require those equal angles. Transform and trench junctions depend on the allowed relative motions and boundary geometry. Stable and unstable evolution were formulated through plate-velocity geometry in the original triple-junction analysis.^[ref]

These original diagrams show an illustrative flexural response and a symmetric ridge-junction construction; they are not maps or measured Hawaiian profiles.

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.

Take upward deflection as positive, use $k>0$, and write $\Delta \rho ={\rho }_m-{\rho }_c>0$. A positive added load depresses the plate, so $w$ is negative beneath it. Because bending moves the original surface while the load adds thickness, the final topography is $e=s+w$. For the cosine mode the horizontal biharmonic operator multiplies $w$ by $k^4$, givingEliminate $s=e-w$ to obtainThe original loading equation contains ${\rho }_m$, whereas the relation after eliminating the added thickness contains $\Delta \rho$. These must not be interchanged prematurely.

At first order in the relief, represent a displaced density interface by a sinusoidal mass sheet. A sheet of surface-density amplitude $\Sigma$ at depth $z$ produces a downward gravity amplitude $2\pi G\Sigma e^{-kz}$ at the datum. This follows by solving the Fourier-transformed gravitational Poisson equation above and below the sheet: the decaying potential has amplitude $-2\pi G\Sigma e^{-k|z-z_0|}/k$, with the normal derivative jump fixed by the sheet mass.

The top interface contributes ${\rho }_{c}e$; the displaced crust–mantle interface contributes $\Delta \rho w$ at depth $t$. ThusThe second term is negative beneath a positive load: depression replaces denser mantle material by lighter crust. Substituting the deflection-to-topography ratio yieldsThis flexural gravity admittance combines mechanical compensation and attenuation of the buried interface's gravitational signal. It assumes small relief, a flat uniform thin plate, uniform density layers and the stated surface loading, without additional internal density anomalies. Here $t$ is the depth of the crust–mantle interface. It is not automatically the effective elastic thickness $T_e$; the latter enters through $D=E{T}_e^3/[12(1-{\nu }^2)]$.

New oceanic lithosphere is hot and mechanically weak. As it cools, the depth over which rock can sustain elastic stress on the loading timescale generally increases, so effective elastic thickness tends to increase with plate age at the time of loading. It is this mechanical state when the load is emplaced, together with subsequent relaxation, that matters; the present age alone is insufficient for an ancient load.

The half-space cooling model illustrates the thermal dependence. With age $\tau$, surface temperature $T_s$, initial interior temperature $T_m$ and constant thermal diffusivity $\kappa$,A chosen temperature $T_{\ast }$ lies atIf the base of the mechanically strong layer approximately follows such an isotherm, $T_e$ initially follows a similar age trend. Cooling models constrained by ocean-floor depth and heat flow provide the thermal motivation; their fitted thermal thickness must not be mistaken for measured elastic thickness.^[ref]

In practice $T_e$ is an effective bending thickness rather than the depth of one universal temperature. Stress-dependent yielding, grain size, water content, composition, loading duration and reheating can alter it. Finite-plate cooling and basal heat input also limit indefinite square-root growth at old ages. Nor should $T_e$ be identified with either the thickness of the crust or the full seismically defined lithosphere.

The planetary crust is the outer compositional rock layer of a differentiated rocky planet. Its thickness need not equal either the lithosphere's thermal thickness or its effective elastic thickness.